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EPRINT  AND  CIRCULAR  SERIES 

OF  THE 

NATIONAL  RESEARCH 
COUNCIL 

MOMENTS  AND  STRESSES  IN  SLABS 

By  H.  M.  Westergaard 

Assistant  Professor  of  Theoretical  and  Applied  Mechanics 
University  of  Illinois 

and 

W.  A.  Slater 
Engineer  Physicist,  U.  S.  Bureau  of  Standards 


V 


Reprinted  from  Proceedings  of  the  American  Concrete  Institute,  vol.  17,  1921 
By  permission  of  the  American  Concrete  Institute 


MOMENTS  AND  STRESSES   IN  SLABS. 
BY  H.  M.  WESTERGAARD  *  AND  W.  A.  Si- 

I.— INTRODUCTION. 

1.  The  subject  of  the  strength  of  flat  slabs  has  received  considerable 
attention  during  the  past  ten  years.     In  November,  1910,  the  floor  of  the 
Deere  and  Webber  Building  J  at  Minneapolis  was  tested.     This  was  the 
first  field  test  of  a  reinforced-concrete  building  floor  in  which  strain  meas- 
urements in  the  reinforcement  and  in  the  concrete  were  taken  at  various 
places  in  the  building.     Since  that  time  many  other  tests  have  been  made 
and  much  study  has  been  given  to  the  analytical  side  of  the  problem. 

While  considerable  work  has  been  done  on  the  correlation  of  the 
analytical  and  the  experimental  results,  it  does  not  seem  that  the  possibili- 
ties of  useful  work  in  this  direction  have  been  exhausted.  It  is  the  purpose 
of  this  paper  to  present  information  which  correlates  the  results  of  tests  of 
a  fairly  large  number  of  slab  structures  with  the  results  of  analysis,  so  that 
the  report  may  aid  in  the  formulation  of  building  regulations  for  slabs. 

The  field  of  this  report  may  be  divided  into  three  parts:  (a)  analysis 
of  moments  and  stresses  in  slabs,  (b)  study  of  the  relation  between  the 
observed  and  the  computed  steel  stresses  in  reinforced-concrete  beams,  made 
for  the  purpose  of  assisting  in  the  interpretation  of  slab  tests,  (c)  a  study 
of  the  test  results  for  flat  slabs  with  a  view  of  comparing  xthe  moments  of 
the  observed  steel  stresses  with  the  bending  moments  indicated  by  the 
analysis,  and  of  estimating  the  factor  of  safety. 

The  mathematical  analysis  is  the  work  of  Mr.  Westergaard.  The 
analysis  of  the  beam  tests  to  show  the  relation  between  the  computed  and 
the  observed  stresses  is  the  work  of  Mr.  Slater. 

2.  ACKNOWLEDGMENT.    The  expense  of  the  report  has  been  borne  jointly 
by   the  American   Concrete   Institute   and   the   United   States   Bureau   of 
Standards. 

The  Corrugated  Bar  Co.,  of  Buffalo,  N.  Y.,  and  A.  R.  Lord,  of  the 
Lord  Engineering  Company,  of  Chicago,  have  furnished  the  results  of  a 
number  of  tests  which  had  not  been  published,  or  which  had  been  published 
only  in  part. 

Acknowledgment  is  made  to-M.  C.  Niehols,  graduate  student,  and  to  J. 
P.  Lawlor  and  K.  H.  Siecke,  seniors  in  engineering,  in  the  University  of 
Illinois,  for  their  assistance  in  working  up  the  data  of  the  tests. 


*  Assistant  Professor  of  Theoretical  and  Applied  Mechanics,  University  of  Illinois, 
t  Engineer  Physicist,  U.  S.   Bureau  of  Standards. 

t  A.    R.    Lord,    Test    of   a    Flat    Slab    Floor    in    a    Reinforced-concrete    Building, 
National  Association  of  Cement  Users,  v.  6,  1910. 


II.— ANALYSIS   OF    HOMOGENEOUS    ELASTIC    PLATES. 

BY  H.  M.  WESTERGAARD. 

3.  SCOPE  OF  THE  ANALYSIS.  A  slab  is  sometimes  analyzed  by  considering 
it  as  divided  into  strips,  each  carrying  a  certain  portion  of  the  total  load. 
One  may  expect  to  obtain,  by  this  method,  an  exact  analysis  of  a  structure 
consisting  of  strips  which  cross  one  another  and  carry  the  loads  as  assumed. 
This  structure,  however,  is  quite  different  from  the  slab.  The  degree  of 
approximation  obtained  may  be  judged  by  the  resemblance  or  lack  of 
resemblance  between  the  strip-structure  and  the  actual  slab.  As  the  resem- 
•  blance  is  not  very  close,  the  approximation,  naturally,  is  not  very  satisfac- 
tory. The  ordinary  theory  of  beams,  too,  is  approximate,  not  exact,  when 
applied  to  actual  beams.  Assumptions  are  introduced  in  the  beam  theory: 
for  example,  the  plane  cross-sections  remain  plane  after  the  bending,  and 
the  material  is  perfectly  elastic.  But  the  approximation  in  the  beam  theory 
is  much  closer  than  in  the  strip  analysis  of  plates.  The  explanation  is 
simple:  the  beam  to  which  the  beam  theory  applies  exactly  has  a  closer 
resemblance  to  actual  beams  than  the  strip-structure  has  to  slabs.  It  is 
possible,  however,  to  analyze  slabs  more  exactly  than  can  be  done  by  the 
strip  method.  If  an  analysis  of  slabs  is  to  compare  in  exactness  with  beam 
analysis,  then  it  must  be  based  on  a  structure  which  resembles  actual  slabs 
more  closely  than  does  the  strip  structure.  It  is  hardly  possible  at  present 
to  cover  by  analysis  the  whole  range  of  designs  of  reinforced-concrete  slabs. 
It  is  expedient,  therefore,  to  confine  this  investigation  to  a  single  type.  The 
homogeneous  slab  of  perfectly  elastic  material  is  selected ;  homogeneous 
slabs  have  a  fairly  close  resemblance  to  other  slabs,  and  exact  methods 
exist  by  which  they  may  be  analyzed.  The  selection  of  a  homogeneous 
elastic  material  agrees  witli  common  practice  in  the  investigation  of 
statically  indeterminate  structures.  For  example,  the  distribution  of  bend- 
ing moments  in  a  reinforced-concrete  arch  or  frame  is  often  determined  by 
replacing  the  structure  by  one  of  homogeneous  material.  The  plan  is  then 
to  investigate  distributions  of  moments  in  homogeneous  slabs.  These  distri- 
butions mav  be  used  as  a  basis  for  the  study  of  the  experimental  data. 

Theoretical  analysis  is  under  the  disadvantage  that  its  processes  are 
often  rather  more  remote  from  the  actual  phenomena  which  are  studied, 
than  are  the  processes  of  direct  physical  tests.  For  this  reason  alone  it 
would  be  out  of  the  question  to  rely  on  the  results  of  theoretical  analysis 
only.  There  are,  on  the  other  hand,  advantages  of  theoretical  analysis 
which  fully  warrant  its  extensive  use  in  conjunction  with  physical  tests. 
One  may  appreciate  these  advantages  by  looking  upon  the  theoretical  analy- 
sis as  being,  in  a  sense,  a  test  in  which  the  testing  apparatus  consists  of 
the  principles  of  statics  and  geometry,  expressed  in  equations,  and  in  which 
the  structure  tested  is  the  structure  assumed  in  the  analysis.  The  equations 
may  be  solved  with  any  desired  exactness,  and  the  structure  has  dimensions 


MOMENTS  AND  STRESSES  IN  SLABS.  3 

and  properties  exactly  as  assumed,  and  is  not  subject  to  the  incidental 
variations  which  so  often  have  made  it  impossible  to  draw  definite  conclu- 
sions from  physical  tests.  Besides,  by  the  analysis  one  may  cover  whole 
ranges  of  variations  of  the  structure  or  at  least  a  great  number  of  indi- 
vidual structures,  while  a  physical  test  can  deal  with  only  a  limited  number 
of  cases.  For  these  reasons  the  analysis  is  of  particular  value  as  a  basis  of 
comparison  and  as  a  method  of  establishing  continuity  between  results  of 
separate,  individual  tests. 

It  will  be  seen  from  the  historical  summary  which  follows  that  the 
problem  of  flexure  of  plates  is  one  on  which  scientists  have  been  at  work 
for  more  than  a  century.  The  methods,  which  have  been  developed  by  such 
men  as  Navier,  St.  Venant,  Kirchhoff,  and  Lord  Kelvin,  have  found  their 
way  into  engineering  literature  in  Europe.  It  has  been  possible,  therefore, 
to  build  the  present  report,  in  part,  on  the  work  of  previous  investigators. 
The  agreement  between  the  results  of  different  analyses  of  rectangular 
slabs  supported  oh  four  sides,  serves  as  evidence  that  the  methods  are 
dependable. 

It  has  been  thought  desirable  to  follow  the  method  of  presenting  the 
results  first,  and  details  of  the  processes  afterwards.  A  historical  sum- 
mary, a  statement  of  the  limitations  of  the  theory,  and  a  derivation  of  the 
fundamental  equations  are  followed  by  the  report  of  results.  The  results 
deal  with  rectangular  slabs  supported  on  four  edges,  and  with  flat  slabs 
supported  on  round  column  capitals.  Details  of  the  analysis  will  be  given 
in  the  appendix  A. 

4.  HISTOBICAL  SUMMARY.  The  incentive  to  the  earliest  studies  of  the 
flexure  of  plates  appears  to  have  been  an  interest  in  their  vibrations,  in 
particular  those  producing  sound,  rather  than  an  interest  in  the  stresses 
and  strength.  Euler,  after  having  developed  his  theory  of  the  flexure  of 
beams,  attempted  to  explain  the  tone-producing  vibrations  of  bells  by 
assuming  a  division  into  narrow  strips  (or  rings),  each  of  which  would 
act  as  a  beam,1  but  this  application  of  the  strip  method  was  not  satisfac- 
tory. Jacques  Bernouilli  (the  younger),  in  a  paper  presented  in  1788,2 
treated  a  square  plate  as  if  it  consisted  of  two  systems  of  crossing  beams  or 
strips,  and  he  attempted  in  this  way  to  explain  the  results  of  Chladni's 
experiments  with  vibrating  plates,3  in  particular  the  so-called  nodal  figures. 
As  might  be  expected,  the  results  of  this  theory  did  not  agree  very  well 
with  the  experimental  data.  In  1809  the  French  Institut,  at  the  instiga- 
tion of  Napoleon,  proposed  as  a  prize  subject  a  theoretical  analysis  of  the 
tones  of  a  vibrating  plate.  Mile.  Sophie  Germain  4  made  some  unsuccessful 
attempts  to  win  this  prize,  but  won  it  in  1815,  when  she  arrived  finally  at  a 


1  Euler,  De  sono  campanarum,  Novi  Commentarii  Academiae  Petropolitanae,  v.   10, 
1766. 

2  Jacques    Bernouilli,    Esaai    theoretique    sur    les    vibrations    des    plaques    elastiques 
rectangulaires     et    libres,     Nova    Acta     Academiae     Scientarum     Petropolitanae,     v.     5, 
1787    (printed   1789). 

*  E.    F.    F.    Chladni,    Entdeckungen    iiber   die   Theorie    des    Klanges,    Leipzig,    1787. 
4  See  Todhunter  and   Pearson,   A   History  of  the  Theory  of  Elasticity,   Cambridge, 
1886,  p.   147. 


4  MOMENTS  AND  STRESSES  IN  SLABS. 

fairly  satisfactory,  though  not  faultless  derivation  of  a  fundamental  equa- 
tion for  the  flexural  vibrations.  But  in  the  meantime,  in  1811,  Lagrange, 
who  was  a  member  of  the  committee  to  pass  on  the  papers,  had  indicated 
in  a  letter  this  equation,  which  is  known,  therefore,  as  Lagrange's  equation 
for  the  flexure  and  the  vibration  of  plates  (with  the  term  depending  on  the 
motion  omitted,  it  is  the  same  as  (11)  in  Art.  6). 

In  1820,  Navier,5  in  a  paper  presented  before  the  French  Academy, 
solved  Lagrange's  equation  for  the  case  of  a  rectangular  plate  with  simply 
supported  edges.  By  this  solution  one  may  compute  the  deflections  and, 
therefore,  also  the  curvatures  and  the  stresses  at  any  point  of  a  plate  of 
this  kind,  under  any  distributed  uniform  or  non-uniform  load.  Navier's 
solution  could  be  applied  only  to  plates  of  this  particular  shape  and  with 
this  type  of  support.  Furthermore,  a  really  acceptable  derivation  of 
Lagrange's  equation,  a  derivation  based  on  the  stresses  and  deformations 
at  all  points  of  the  plate,  had  not  been  found  so  far.  Poisson,6  in  his  famous 
paper  on  elasticity,  published  in  1829,  obtained  such  a  proof.  With  it,  he 
derived  a  set  of  general  boundary  conditions  (conditions  of  equilibrium 
and  of  deformation  at  the  edge  of  the  plate),  and  was  then  able  to  obtain 
solutions  for  circular  plates,  both  for  vibrations  and  for  static  flexure  under 
a  load  which  is  symmetrical  with  respect  to  the  center.  Poisson's  theoret- 
ical results  were  compared  with  results  of  tests,  namely,  with  the  experi- 
mental values,  found  by  Savart  for  the  radii  of  the  nodal  circles  of  three 
vibrating  circular  plates.  A  close  agreement  was  found. 

In  a  paper,  published  1850,  Kirchhoff7  derived  Lagrange's  equation  and 
the  corresponding  boundary  conditions  by  using  the  energy  principle,  or  the 
principle  of  least  action.  He  found  one  boundary  condition  less  than 
Poisson,  namely,  four  at  each  point  instead  of  Poisson's  five.  This  differ- 
ence gave  rise  to  some  discussion,  but  finally,  in  1867,  Kelvin  and  Tait ' 
showed  that  there  was  only  an  apparent  discrepancy,  due  to  an  inter- 
relation between  two  of  Poisson's  conditions.  This  conclusion,  as  well  as 
Kirchhoff's  and  Poisson's  theories  as  a  whole,  applies,  as  might  be  expected, 
with  limitations  which  are  analogous  to  the  limitations  of  the  ordinary 
theory  of  beams.  For  example,  the  plate-theory  ceases  to  apply  when  the 
span  becomes  small  compared  with  the  thickness  of  the  plate,  but,  of  course, 
in  that  case  the  structure  has  really  ceased  to  be  a  plate  in  the  ordinary 
sense.  The  question  of  the  exact  nature  of  the  limitations  called  for 
further  researches.  Such  were  made  by  Bouissinesq.9  His  investigations 
have  established  the  applicability  of  Poisson's  and  Kirchhoff's  theories  to 


*  See    Saint-Venants   annotated   edition   of   Clebsch's   Theory   of    Elasticity,    Paris, 
1883.      Note  by   Saint-Vcnant,   pp.    740-752. 

*  S.    D.    Poisson,    Memoire    sur    1'equilibre    et    le    mouvement    des    corps   elastique, 
Memoirs  of  the  Paris  Academy,  v.  8,  1829,  pp.  357-570.     See  Todhunter  and  Pearson, 
History  of  the  Theory  of  Elasticity,   1886,  pp.  241,  272. 

*  G.    Kirchhoff,    Ueber    das    Gleichgewicht    und    die    Bewegung    einer    elastischen 
Scheibe,  Crelles  Journal,  1850,  v.  40,  pp.  51-88. 

1  Kelvin  and  Tait,  Natural  Philosophy,  ed.  1,  1867.  See  A.  E.  H.  Love,  Mathe- 
matical Theory  of  Elasticity,  ed.  1906.  p.  438. 

6  J.  Boussinesq,  fitude  nouvelle  sur  1'equilibre  et  le  mouvement  des  corps  solides 
elastiq'ues  dont  certaines  dimensions,  sont  tres-petites  par  rapport  a  d'autres,  Journal 
de  Mathematiques,  1871,  pp.  125-274.  and  1879,  pp.  329-344. 


MOMENTS  AND  STRESSES  IN  SLABS.  5 

homogeneous  elastic  plates  whose  ratio  of  the  thickness  to  the  span  is 
neither  very  large  nor  very  small,  that  is,  plates  whose  dimensions  are  not 
extreme. 

With  a  theoretical  foundation  thus  laid,  the  time  was  ready  for  efforts 
to  obtain  numerical  results  by  application  of  the  theory,  that  is,  by  solution 
of  the  general  differential  equation  in  specific  cases  of  technical,  or  other- 
wise scientific  importance.  There  was  due  also  a  change  of  chief  interest 
in  the  problem  from  the  question  of  vibrations  to  that  of  stresses  and 
strength,  that  is,  the  time  had  come  for  the  structural,  rather  than  the 
acoustic  problem  to  stand  in  the  foreground.  Lavoinne,10  in  1872,  tackled 
the  question  of  a  plane  boiler  bottom  supported  by  stay-bolts.  The  problem 
is  essentially  the  same  as  that  of  the  flat  slab  (of  homogeneous  material) 
supported  directly  on  column  capitals,  without  girders,  and  carrying  a 
uniform  load.  Lavoinne's  solution  is  for  the  case  in  which  Poisson's  ratio 
of  lateral  contraction  is  equal  to  zero,  but,  as  will  be  shown  later,  a  correc- 
tion for  this  lateral  effect  may  be  made  afterward  without  any  difficulty. 
Lavoinne,  by  the  use  of  a  double-infinite  Fourier  series,  solves  Lagrange's 
equation  for  a  uniformly  loaded,  infinitely  large  plate  which  is  divided  by 
the  supports  into  rectangular  panels,  and  which  has  its  supporting  forces 
uniformly  distributed  within  small  rectangular  areas  around  the  corners 
of  the  panels.  The  scries  for  the  load  become  divergent  when  the  size  of 
the  rectangles  of  the  supporting  forces  becomes  zero,  that  is,  when  the 
supports  are  point-supports.  The  same  problem  was  treated  by  Grashoff," 
whose  solution,  however,  is  incorrect,  since  it  disregards  some  of  the 
boundary  conditions.  G.  IT.  Bryan,12  in  1890,  made  an  analysis  of  the 
buckling  of  a  rectangular  elastic  plate,  due  to  forces  in  its  own  plane. 
Maurice  Levy 13  showed  how  Lagrange's  equation,  when  applied  to  rec- 
tangular plates  with  various  types  of  supports,  may  be  integrated  by  a 
single-infinite  series  depending  on  hyperbolic  functions,  instead  of  the 
double-infinite  Fourier  series  in  Navier's  solution. 

In  the  meantime,  a  different  path  of  investigation,  namely,  that  of 
semi-empirical  methods,  had  been  entered  into  by  Galliot  and  by  C.  Bach. 
Galliot"  compared  observed  deflections  of  plates  in  lock-gates  (under 
hydraulic  pressure)  with  the  results  of  an  approximate  theory,  which,  in 
this  manner,  he  found  applicable  as  a  basis  of  design.  Bach's  15  empirical 
formulas  are  based  on  laboratory  tests  in  connection  with  some  very  simple 
theoretical  considerations.  An  example  will  illustrate  his  method.  He 
found  by  test  that  the  line  of  failure,  the  danger  section,  of  a  square  plate, 


10  Lavoinne,  Sur  la  resistance  des  parois  planes  des  chaudieres  a  vapeur,  Annales 
des  Fonts  et  Chaussees,  v.  3,  1872,  pp.  276-303. 

11  F.  Grashof,  Elasticitat  und  Fcstigkeit,  ed.  1878,  p.  351. 

13  G.  H.  Bryan,  On  the  stability  of  a  plane  plate  under  thrusts  in  its  own  plane, 
London  Math.   Soc.,  v.  22,  1890,  pp.  54-67.  _ 

18  Maurice    Levy,    Sur    1'equilibre    elastique    d'une    plaque    rectangulaire,    Comptes 
Rendus.  v.  129,  1899,  pp.  S3S-539. 

14  Galliot,  fitude  sur  les  portes  d'ecluses  en  tole,  Annales  des  Fonts  et  Chaussees, 
1887,  v.  14,  pp.  704-756. 

16  C.   Bach,  Versuche  fiber  die  Widerstandsfahigkeit  ebener  Flatten,   Zeitschr.     d. 
Ver.  deutscher  Ingenieure,  v.    34,   1890,  pp.   1041-1048,   1080-1086,    1103-1111,1139-1144. 


6  MOMENTS  AND  STRESSES  IN  SLABS. 

simply  supported  along  the  edges,  is  along  the  diagonals.  It  happens  that 
the  average  bending  moment  per  unit  length  across  the  diagonal  can  be 
determined  by  a  simple  analysis  based  on  elementary  principles  of  statics 
(the  result  is  1/24  id2  where  w  is  the  load  per  unit-area,  I  the  span).  But 
this  analysis  gives  no  information  as  to  the  distribution  of  the  bending 
moment.  Bach,  then,  multiplies  the  average  moment  by  an  empirical  con- 
stant, found  by  comparison  with  the  tests.  The  investigation  included  cases 
of  rectangular  and  circular  plates,  with  distributed  or  concentrated  loads. 
Bach's  formulas,  because  of  their  simplicity  and  sound  empirical  basis, 
have  been  used  rather  extensively.  A  similar  treatment  of  the  problem 
of  the  plane  boiler-bottom,  supported  by  stay-bolts,  was  added  later.16  The 
analysis  of  flat  slabs,  which  was  indicated  in  1914  by  Nichols,"  may  be 
recognized  as  falling  into  the  same  category  as  Bach's  analyses.  Since  its 
first  appearance,  Nichol's  analysis  has  been  used  by  many  as  a  basis  of 
comparison  between  results  of  tests  and  rules  of  design. 

Since  the  close  of  the  nineteenth  century  the  investigators  of  the  theo- 
retical side  of  the  question  have  been  confronted  with  three  definite  tasks. 
Analyses  which  would  cover  the  extreme  cases  in  which  the  plate  is  either 
very  thick  or  very  thin  were  called  for;  new  theoretical  methods  were 
needed,  for  example,  for  the  solution  of  Lagrange's  equation;  and  numer- 
ical results  applying  to  specific  cases  had  to  be  worked  out. 

Theories  applying  to  plates  of  ordinary  thickness,  as  well  as  to  thick 
plates  with  a  short  span,  have  been  developed  by  Michell,18  Love,19  and 
Dougal."0  The  latter,  when  he  applied  the  exact  theory  to  plates  of  ordi- 
nary thickness,  found  agreement,  in  a  number  of  specific  cases,  with  the 
results  derived  by  Lagrange's  equation.  Thin  plates,  whose  deflections 
have  become  so  large  compared  with  the  thickness  that  the  curving  of  the 
cross-section  must  be  considered,  have  been  treated  by  A.  Foppl.-1 

As  an  example  of  the  development  of  methods,  mention  may  be  made 
again  of  Levy's  solution  of  hyperbolic  functions.  Dougall.  in  the  paper 
just  quoted,  used  Bessel-functions,  and  obtained  thereby  some  rapidly  con- 
verging solutions.  Other  solutions  by  various  series  have  been  contributed 
by  Hadamard,"  Lauricella,"3  Happel,  24  and  Botasso.25  The  modern  theory 


16  C.    Bach,    Die    Berechnung   flacher,   durch    Anker   oder    Stehbolzen    unterstiitzer 
Kesselwandungen    und    die    Ergebnisse    der    neuesten    hierauf    bezuglichen    Versuche, 
Zeitschr.   d.   Ver.   deutscher   Ingenieure,    1894,   pp.    341-349. 

17  T.    R.    Nichols,    Statical    limitations    upon    the    steel    requirement    in    reinforced- 
concrete  flat  slab  floors.  Am.   Soc.  C.  E.,  Trans.,  v.   77.  1914,  pp.   1670-1681. 

18  J.    H.    Michell,    On    the   direct   determination    of   stress   in   an   elastic   solid,    with 
application  to  the  theory  of  plates,  London  Math.   Soc.   Proc.,  v.  31,   1899,  pp.   100-124. 

19  A.   E.   H.   Love,  Mathematical  Theory  of  Elasticity,   ed.   1906,  pp.   434-465. 

20  J.  Dougall,  An  analytical  theory  of  the  equilibrium  of  an  isotropic  elastic  plate, 
Edinburgh   Royal   Soc.   Trans,  v.   41,   1903-4.  pp.   129-227. 

"A.  Foppl,  Technische  Mechanik.  v.  5,  ed.  1918,  pp.  132-144;  also:  A.  and  L. 
Foppl,  Drang  und  Zwan?,  v.  1.  1920,  pp.  216-232. 

22  J.  Hadamard,  Stir  le  probleme  d'  analyse  relatif  a  1'equilibre  des  plaques 
elastiqucs  encastrces.  Institut  de  France,  Acad.  des  Sciences,  Memoires  presentes  par 
divers  savants,  v.  33,  1908,  Xp.  4,  128  p.p. 

21 G.  Lauricella,  Sur  1'integration  de  1'cquation  relative  a  1'cquilibre  des  plaques 
elastiques  encastrees,  Acta  Mathematica,  v.  32,  1909,  pp.  201-256. 

-*  H.  Happel,  Ueber  das  Gleichgewicht  rechteckiger  Flatten.  Gottinger  Nachrichten, 
Math.  phys.  Klasse,  1914,  pp.  37-62  (rectangular  plate  with  fixed  edges  and  with  a 
concentrated  load  at  the  center). 

25  Matteo  Botasso,  Sull'equilibrio  delle  piastre  elastiche  piane  appoggiate  lungo  il 
contorno,  R.  Accademia  della  scienze  di  Torino,  Atti,  v.  50,  1915,  pp.  823-838. 


MOMENTS  AND  STRESSES  IN  SLABS.  7 

of  integral  equations  has  opened  the  way  for  new  solutions,  by  series  which 
may  fit  almost  any  type  of  plate  (see,  for  example,  Hadamard's  and 
Happel's  \vorks,  which  were  just  quoted).  A  method  of  a  different  type  is 
Ritz's  28  approximate  method,  which  was  indicated  in  1909,  which  may  be 
applied  to  any  elastic  structure,  and  which  was  applied  by  Ritz  himself  to 
plate  problems,  and  after  him,  by  other  writers,  to  water  tanks,  domes,  etc., 
and  to  plates.  The  method  makes  use  of  series  of  properly  chosen  functions, 
each  of  which  must  satisfy  the  boundary  conditions  of  the  problem,  and 
each  of  which  is  introduced  in  the  series  with  a  variable  coefficient  which  is 
unknown  beforehand.  Then  one  determines  a  suitable,  finite  number  of 
these  coefficients  by  the  principle  of  energy-minimum.  Ritz's  method  has 
proved  itself  an  effective  addition  to  our  analytical  equipment.  Another 
approximate  method  is  that  of  difference  equations  which  was  used  by  N.  J. 
Nielsen"  in  a  work  on  stresses  in  plates.  His  results  will  be  mentioned 
later.  By  the  method,  the  differentials  of  the  differential  equations  are 
replaced  by  finite  differences,  and  the  problem  is  then  reduced  to  the  solution 
of  a  set  of  linear  equations,  in  which,  for  example,  the  deflections  at  a  finite 
number  of  points  enter  as  variables.  The  method  is  used,  in  fact,  when 
string  curves  for  distributed  loads  (for  example,  in  the  investigation  of 
beams)  are  replaced  by  string  polygons. 

Investigations  in  the  theory  of  plates,  made  with  the  purpose  of  obtain- 
ing definite  results  in  specific  cases,  have  appeared  in  a  fairly  great  number 
during  recent  years.  Estanave,28  in  a  thesis  in  Paris,  1900,  analyzed  various 
cases  of  the  flexure  of  rectangular  plates.  Simic,29  in  1908,  gave  an  approxi- 
mate solution  for  rectangular  plates  with  simply  supported  edges.  He 
used  a  rather  short  series  of  polynomials.  The  results  agree  fairly  well 
with  those  found  by  later  investigations.  Hager,30  in  a  work  published 
in  1911,  applied  trigonometric  series,  and  used  Ritz's  method,  in  an  investi- 
gation of  rectangular  slabs.  His  results  are  incorrect,  in  so  far  as  they 
apply  to  homogeneous  plates,  because  the  torsional  moments  in  the  sections 
parallel  to  the  edges  are  not  considered;  the  results  may,  however,  have 
some  interest  with  reference  to  two-way-reinforced  concrete  slabs,  which 
have  a  reduced  torsional  resistance  in  these  sections.  The  same  criticism 
applies  to  an  investigation,  first  published  in  1911,  by  Danusso.31.  He 


28  Walter  Ritz,  Ueber  cine  neue  Methode  zur  Losung  gewisser  Variationsprobleme 
der  mathematischen  Physik,  Crelles  Journal,  v.  135,  1909,  pp.  1-61.  See  also  H. 
Lorcnz,  Technische  Elastizitatslehre,  1913,  p.  397. 

27  N.   J.    Nielsen,   Bestemmelse   af   Spaendinger   i    Plader   ved   Anvendelse   af   Dif- 
ferensligninger,  Copenhagen,   1920. 

28  E.    Estanave,   Contribution   a  1'etude  de  1'equilibre  elastique  d'une  plaque  mince, 
Paris,  1900. 

29  Jovo  Simic,  Ein  Beitrag  zur  Berechnung  der  rechteckigen  Flatten,  Zeitschr.  des 
oesterr.     Ingenieur-  und  Architekten-Vereines,  v.  60,  1908,  pp.  709-714.     Another  paper 
by   Simic    (Oesterr.      Wochenschrift   fur   den   offentlichen    Baudienst,    1909),    was   criti- 
cized by  Mesnager   (see  the  paper  quoted  later,   of  1916,  p.  417)   on  the  ground  that 
torsional  moments  had  not  been  duly  considered. 

30  Karl    Hager,    Berechnung   ebener    rechteckiger    Flatten    mittels    trigonometrischer 
Reihen,  Munich  and  Berlin,   1911.     For  criticism,  see  Mesnager's  paper  of  1916,  which 
is  quoted  below,  pp.  414-418. 

31  See  Arturo   Danusso,    Beitrag   zur  Berechnung  der  kreuzweise  bewehrten   Eisen- 
betonplatten  und  deren  Aufnahtnetrager.     Prepared  in  German  by  Hugo  von  Bronneck 
after  the  articles  by  Danusso  in  II  Cemento,  1911,  No.  1-10;    Forscherarbeiten  auf  dem 
Gebiete  dcs  Eisenbetons,  v.  21,   1913,   114  pp. 


8  MOMENTS  AND  STRESSES  IN  SLABS. 

replaces  the  rectangular  slab,  as  Jacques  Bernoulli!  had  done  in  1789,  by 
two  systems  of  crossing  beams  which  are  connected  at  the  points  of  inter- 
section, only  he  considers  a  finite,  instead  of  an  infinite  number  of  such 
beams.  This  structure,  again,  has  no  torsional  resistance  in  the  sections 
parallel  to  the  sides.  A  structure  consisting  of  three  closely  spaced  systems 
of  crossing  beams  in  three  different  directions  would,  on  the  other  hand, 
have  both  torsional  and  bending  resistance  in  all  directions.  Such  a  struc- 
ture was  used  by  Danusso  in  the  analysis  of  a  triangular  plate,  and  his 
results  may  be  expected  to  be  approximately  correct  in  this  special  case,  as 
long  as  Poisson's  ratio  may  be  assumed  equal  to  zero. 

In  a  note  issued  in  1912  by  the  French  Council  on  Bridges  and  Roads12 
some  design  formulas  were  presented,  together  with  various  analyses  based 
on  the  differential  equation  of  flexure.  The  years  1913  to  1916  brought 
forth  a  rather  valuable  collection  of  exact  or  approximately  exact  studies 
of  rectangular  slabs  supported  on  four  sides.  The  authors  referred  to  are 
Hencky,"  Paschoud,34  Leitz,35  Xadai,34  and  Mesnager,37  and  they  appear  to 
have  worked  entirely  independently  of  each  other.  Their  numerical  results 
agree,  on  the  whole,  very  well.  A  treatment  of  flat  slabs  by  Eddy,18  pub- 
lished in  1913,  was,  unfortunately,  not  free  from  faults.  Incorrect  boundary 
conditions,  inconsistencies  in  the  consideration  of  the  negative  moments 
across  the  rectangular  belts,  and  the  use  of  an  abnormally  high  value  of 
Poisson's  ratio,  namely,  one-half,  led,  naturally,  to  incorrect  results.  One 
may  also  object  to  his  use  of  the  terms  "true"  and  apparent"  stresses  and 
bending  moments  in  a  manner  which  is  contrary  to  common  usage. 

N".  J.  Nielsen's27  investigation,  published  in  1920,  was  mentioned  on 
account  of  the  use  of  difference  equations.  He  proved  the  applicability  of 
the  method  by  applying  it  to  known  cases,  where  the  results  found  by 
previous  investigators  had  shown  approximate  agreement.  Analyses  of 
rectangular  slabs  supported  on  four  sides  served  this  purpose.  He  then 
analyzed  the  action  of  flat  slabs  with  different  loading  arrangements,  with 
square  or  rectangular  panels,  stiff  or  flexible  columns,  etc.,  and  he  made 
special  studies  of  the  stresses  in  exterior  panels  and  corner  panels.  The 
approximation  obtained  does  not  seem  to  be  quite  satisfactory  in  all  the 


12  Conceil  General  des  Fonts  et  Chaussees,  Calcul  des  hourdis  en  beton  arme, 
Annales  des  Fonts  and  Chaussees,  1912,  VI,  pp.  469-529. 

33  H.  Hencky,  Ueber  den  Spannungszustand  in  rechteckigen  ebener  Flatten,  1913, 
94  pp.  (thesis  in  Darmstadt). 

*4  Maurice  Paschoud,  Sur  I'applicatipn  de  la  methode  de  Walter  Ritz  a  1'etude  de 
1'equilibre  elastique  d'une  plaque  carree  mince,  thesis  in  Paris,  1914,  56  pp.  (See 
Mesnager's  paper,  quoted  later.) 

**  H.  Leitz,  Die  Berechnung  der  frei  aufliegenden,  rechteckingen  Flatten,  Forsch- 
erarbeiten  auf  dem  Gebiete  des  Eisenbetons,  v.  23,  1914,  59  pp.  He  added  later  an 
analysis  of  rectangular  plates  with  fixed  edges,  see  his  paper:  Die  Berechnung  der 
eingespannten,  rechteckigen  Platte,  Zeitschr.  f.  Math.  u.  Phys.,  v.  64,  1917,  pp.  262-272. 

"  Arpad  Nadai.  Die  Formanderungen  und  die  Spannungen  von  rechteckigen 
elastichen  Flatten,  Forschungsarbeiten  auf  dem  Gebiete  des  Ingenieurwesens,  v.  170- 
171,  1915,  87  pp.  Also,  in  a  shorter  presentation,  in  Zeitschr.  d.  Ver.  deutscher 
Ingenieure,  1914,  .pp.  487-494,  540-550. 

r  Mesnager,  Moments  et  fleches  des  plaques,  rectangulaires  minces,  portant  une 
charge  uniformement  repartie,  Annales  des  Fonts  et  Chaussees.  1916,  IV,  pp.  313-438. 

**  H.  T.  Eddy,  The  theory  of  the  flexure  and  strength  of  rectangular  flat  plates 
applied  to  reinforced-concrete  floor  slabs,  1913. 


MOMENTS  AND  STRESSES  IN  SLABS.  9 

cases.  This  deficiency  might  have  been  remedied  by  the  use  of  a  greater 
number  of  terms,  but  thereby  the  complexity  of  the  work  would,  of  course, 
have  increased.  Nielsen's  analysis  is  the  first  in  which  approximately  exact 
methods  of  analysis  are  applied,  on  an  extensive  scale,  to  the  flat-slab  prob- 
lem. His  results  will  be  quoted  later,  on  various  occasions. 

The  experimental  work  on  steel  plates  had  been  continued,  in  the 
meantime,  by  Bach."1  Another  contribution  of  the  sort  is  due  to  Craw- 
ford.40 A  test  of  a  rubber  model,  designed  to  represent  a  flat-slab  structure, 
was  made  by  Trelease  tt  for  the  Corrugated  Bar  Co.  The  experimental  work 
on  concrete  slabs  is  mentioned  at  other  places  in  this  report. 

Among  the  treatises  in  which  the  slab-problem  is  dealt  with  extensively 
may  be  mentioned  those  by  Love,  Foppl,  and  Lorenz.42 

5.  LIMITATIONS  OP  THE  THEORY.     The  properties  of  the  plates  dealt 
with  in  the  following  analysis  will  now  be  defined. 

a.  In  order  to  simplify  the  discussions  the  plates  will  be  assumed  to  be 
horizontal,  the  applied  forces  vertical. 

6.  The  plates  are  medium-thick.    A  medium-thick  plate  is  defined  here 
as  one  which  is  neither  so  thick  in  proportion  to  the  span  that  an  appre- 
ciable portion  of  the  energy  of  deformation  is  contributed  by  the  vertical 
stresses   ( shears,  tensions,  and  compressions ) ,  nor  so  thin  that  an  appre- 
ciable part  of  the  energy  is  due  to  the  stretchings  and  shortenings  of  the 
middle  plane  when  the  plate  is  bent  into  a  double-curved  surface.     All 
plates  and  plate-like  structures  may  be  divided  into  four  groups  according 
to  thickness:    thick  plates,  in  which  the  vertical  stresses  are  important; 
medium-thick  plates,  to  which  the  present  analysis  applied;    thin  plates, 
whose  resistance  to  transverse  loads  depends  in  part  on  the  stretching  of 
the  middle  plane;    and  membranes,  which  are  so  thin  that  the  transverse 
resistance  depends  exclusively  on  the  stretching.    The  membrane,  of  course, 
is  not  a  plate  in  the  ordinary  sense,  any  more  than  a  suspended  cable  is  a 
beam.    The  thick  and  the  thin  plates  require  special  theories,  such  as  those 
developed  by  Michell,  Love,  Dougall,  and  Foppl  (see  Art.  4).    These  extreme 
cases  are  eliminated  by  definition.    Plates  of  such  proportions  as  are  gener- 
ally used  in  reinforced-concrete  floor  slabs  may  be  classified  as  medium- 
thick,  and  fall  within  the  scope  of  the  analysis. 

c.  The  plates  are  homogeneous  and  of  uniform  thickness.  Since  the 
vertical  stresses  do  not  contribute  directly  to  the  work  of  deformation  or  to 
the  deflections,  it  is  sufficient  to  specify  the  elastic  properties  with  regard 


89  C.  Bach.  Versuche  iiber  die  Formanderung  tind  die  Widerstandsfaliigkeit  ebeiier 
Wandungen,  Zeitschr.  d.  Ver.  deutscher  Ingenieure,  1908,  pp.  1781-1789,  1876-1881. 

40  W.   J.   Crawford,  The  elastic  strength  of  flat  plates,   an  experimental   research, 
Edinburgh  Roy.   Soc.   Proc.,  v.   32,   1911-1912,  pp.   348-389.     Additional  note  by  C.   G. 
Knott,  pp.  390-392. 

41  Corrugated  Bar  Co.,  Bulletin  on  flat  slabs,  Buffalo,  1912. 

"A.  E.  H.  Love,  Mathematical  Theory  of  Elasticity,  ed.   1906,  Chapter  XXII. 

A.  Foppl,  Technishe  Mechanik,  v.  3  and  v.  5  (ed.  1919  and  1918). 

A.  and  L.  Foppl,  Drang  und  Zwang,  v.  1,  1920. 

H.  Lorenz,  Technische  Elasticitatslehre,  1913,  Chapter  VII. 

See  also:  Todhunter  and  Pearson,  History  of  the  theory  of  elasticity,  1886-1893; 
and:  Encyclopedic  der  mathematischen  Wissenchaften,  Vol.  IV,  25,  1907,  pp.  181-190, 
and  27,  1910,  pp.  348-352. 


10  MOMENTS  AND  STRESSES  IN  SLABS. 

to  the  horizontal  strains,    llooke's  law  is  assumed  to  apply  to  the  horizontal 
strains,  and  the  elastic  properties,  then,  depend  on  two  constants;    namely, 
E  =  modulus   of   elasticity   for  horizontal   tensions   and   compres- 
sions, and 

K  =  Poisson's  ratio  of  lateral  horizontal  contraction   to  longitu- 
dinal horizontal  elongation. 

d.  A  straight  line,  drawn  vertically  through  the  plate  before  bending, 
remains  straight  after  bending.     This  assumption  is  consistent  with   the 
preceding  specifications  that  the  plate  is  medium-thick,  and  is  homogeneous 
and  of  uniform  thickness,  and  it  is  entirely  analogous  to  the  assumption  in 
the  theory  of  beams  that  a  plane  cross-section  before  bending  remains  plane 
after  bending.     It  follows  that  the  horizontal  unit-stresses,  tensions,  com- 
pressions,   and    shears,    in    vertical    sections    are   distributed    according   to 
straight-line  diagrams,  as  the  tensions  and  compressions  in  the  cross-section 
of  a  beam. 

e.  The  zero-points  in  these  diagrams  for  the  horizontal  stresses  in  ver- 
tical sections  are  in  the  middle  plane,  which  is  therefore  a  neutral  plane. 

As  to  the  question  of  the  consistency  of  these  assumed  properties  refer- 
ence may  be  made  to  the  theoretical  works  mentioned  in  the  historical  sum- 
mary (Art.  4) . 

6.  THE  EQUATIONS  APPLYING  TO  A  SMALL  KECTANGULAB  ELEMENT 
OF  THE  SLAB. 

The  following  notation  is  used: 

x.  y     =  horizontal  rectangular  coordinates  (see  Fig.  1). 
z         =  vertical  deflection,  positive  downward. 

V  =  vertical  shear  per  unit  length  in  the  section  perpendicular  to  a;  at 
noint  (x,  tj)  ;  the  positive  direction  of  V  is  indicated  by  the 
arrow  in  Fig.  1. 

V .      =  same  in  section  perpendicular  to  y. 

A/x      =  bending  moment  per  unit  length  in  the  section  perpendicular  to  a? 
at  point  (x,  y}  ;  M     is  positive  when  causing  compression  at  the 
top  and  tension  at  the  bottom. 
M       =:  same  in  section  perpendicular  to  y. 

M       —  torsional  moment   per  unit  length   in   sections  perpendicular  to  x 
and  ?/  at  point   (x,  y}  ;    the  positive  directions  are  indicated  by 
the  arrows  in  Fig.  1 ;    that  is,  the  torsional  moment  is  considered 
positive  when  it  causes  shortenings  at  the  top  along  the  diagonal 
through  the  corner  (#,?/)   of  the  element. 
K        =  modulus  of  elasticity  of  the  material. 
K        =Poisson's   ratio   of   lateral   contraction   to   longitudinal   elongation. 

Concerning  E  and  K,  see  the  preceding  article. 

7  —moment  of  inertia  per  unit  longlh;  /  =  ^  d3  when  d  is  the  thick- 
ness of  the  slab. 

Fig.  1  shows  a  small  rectangular  element  of  the  slab  with  the  forces 
and  couples  acting  on  it.  The  location  of  the  element  is  denned  by  the 
horizontal  coordinates  x  and  ;/  of  the  midpoint  of  the  lower  left-hand  edge 


MOMENTS  AND  STRESSES  IN  SLABS. 


11 


in  the  figure.  The  dimensions  are  dx  and  dy  in  the  x-  and  y-directions,  and 
the  thickness  of  the  slab  in  the  ^-direction.  The  deflection  at  point  (x,  y)  is 
measured  by  ~,  which  is  positive  downward. 

The  loads  are:    first,  the  applied  surface  load  w  per  unit-area,  in  the 
^-direction;    that  is,  a  total  load  of  w  dx  dy;  secondly,  the  internal  vertical 


y 


K'j^M 


*  surface  had  in  d/recfon  z. 
x  ondy  are  ftorizenfa/ cvordinates. 


cfx 


FIG.  1. — KECTANGULAB  ELEMENT  OF  SLAB. 

shears,  bending  moments  and  torsional  moments  which  are  listed  in  Table  1. 
The  values  per  unit-length  are  y     y    + x  dx   y    etc.,  hence 

the  total  values  are    yx  dy,   (Fx   +  -=-x  dx)   dy,   Vydx >  etc-    The 

ox 

vertical  shears  and  the  bending  moments  are  of  the  same  nature  as  the 
vertical  shears  and  the  bending  moments  in  beams.    The  torsional  moments 


12 


MOMENTS  AND  STRESSES  IN  SLABS. 


M  are  resultants  of  the  horizontal  shears  in  the  vertical  faces.  The  values 
of  M  for  the  lower  and  left-hand  faces  in  Fig.  1  are  equal  on  account  of 
the  law  of  equality  of  shears  in  sections  perpendicular  to  one  another. 

TABLE  I. — FORCES  AND  COUPLES  IN  FIG.   1. 
TABLE    I. 


Face 

Vertical  shear 

Bending  moment 

Torsional  moment 

Value 

Direc- 
tion 

Value 

Direc- 
tion 

Value 

Direc- 
tion 

Left  face 

tidy 

-z 

** 

xz 

Mzdy 

yz 

Riqht  face 

(V^dxty 

+z 

fa<$*tt 

zx 

(M^dxjdy 

zy 

Lower  face  in  F/q.  1. 

Vyd* 

-z 

Mydx 

yz 

Mtdx 

m 

Upper  face  in  F/g.l. 

fa*$W* 

*J* 

(Myt^&dyjdx 

zy 

(M^dy^ 

zx 

Tlie  directions  indicated  are  the  positive  directions  of  the  loads. 

The  forces  shown  in  Fig.  1  must  hold  the  element  of  the  slab  in  equi- 
librium. By  equating  the  sum  of  the  vertical  components  (in  the  z-direc- 
tion)  to  zero  and  dividing  by  dx  dy  we  find 


<5x      by  ,  (1) 

By  equating  to  zero  the  sum  of  the  moments  about  a  line  parallel  to  y 
through  the  center  of  the  element,  and  dividing  by  dx  dy,  we  find 


and  by  analogy, 


Vl"L7        -rr 
T^7      =    VV 


(2) 


(3) 


By  differentiating  (2)  and  (3)  with  respect  to  x  and  y,  respectively, 
and  substituting  in  (1),  we  find 


The  equations  (!)  to  (4)  arc  equations  of  equilibrium.  It  the  slab  is 
bent,  say,  in  the  ^-direction  only,  so  that  the  lines  parallel  to  y  remain 
straight  and  parallel  to  j/,  then  the  slab  acts  as  a  beam,  V  ,  M  ,  and  M  7 
become  zero,  and  the  formulas  (1)  to  (4)  are  reduced  to  the  well-known 
equations  from  beam  theory: 


X        _j 

i      '     x  ~~   x     • 
dx  ox 


j  » 
dxz 


The  terms  containing   M    represent  the  effect  of  the  torsional  resistance. 


MOMENTS  AND  STRESSES  IN  SLABS. 


13 


The  equations  ( 1 )  to  ( 4 )  were  derived  by  the  statical  conditions  of 
equilibrium  without  reference  to  the  deformations.  That  is,  ( 1 )  to  ( 4 ) 
apply  without  reference  to  the  particular  elastic  properties.  Since  they  are 
merely  equations  of  equilibrium,  they  apply  to  non-homogeneous  slabs,  such 
as  reinforced-concrete  slabs,  and  to  slabs  with  reduced  torsional  resistance, 
as  well  as  to  the  homogeneous  elastic  plates. 

We  now  consider  the  deformations  and  their  relations  to  the  loads. 
The  plate  is  again  assumed  to  be  homogeneous.  Fig.  2  illustrates  three 
types  of  deformation:  bending  in  the  a?-direction  shown  in  Fig.  2 (a)  ;  in 
the  t/-direction  shown  in  Fig.  2(b)  ;  and  torsion  in  the  a?i/-directions  shown 
in  Fig.  2(c).  Any  state  of  flexure  of  an  element  of  the  slab  may  be 
resolved  into  component  parts  of  these  three  types.  The  amounts  of  defor- 

827, 

mation  are  measured  in  Fig.  2 (a)   and  Fig.  2(b)  by  the  curvatures  —      — _ 

S2 
an(j ?.       (as  in  beams),  and  in  Fig.  2(c)  by  the  rate  of  change  of  slope, 


that  is,  by     — 


8x 


82 
Z 

—  8\8)/ 


(a)  (bl  '"I 

FIG.  2. — DEFORMATIONS  OF  ELEMENT  OF  SLAB. 


A  bending  moment  M     acting  alone,  produces  the  curvatures    _ —  = 

M  X'  82rc  M  ^X 

— ?     in  the  or-direction,      _   —   _  JT—?    in  the  y-direction,  and  no  twist; 

El  Sy2  El 

these  results  may  be  taken  directly  from  the  theory  of  beams.    The  torsional 


couples  M     produce  a  twist    _ 


Sx 


which  may  be  determined  by  intro- 


ducing  temporarily  another  system  of  coordinates,  as'  ,  y'  ,  making  angles  of 
45°  with  the  system  of  on,  y.  The  couples  M  are  replaced  by  an  equivalent 
combination  of  couples  consisting  of  the  bending  moments  M  —M  in  the 

in  the  i/'-direction.     These  bending  moments 

Mz-(l+K)       82z 
_  —  —    ___  ~ 

El 


x  -direction  and  M     =  — M 

y  z 

82 
produce  the  curvatures 

j    ,2  ~ 
OX 


El  El  8y'  El 

in  terms  of  which  the  twist  may  be  expressed  by  the  transformation  formula 


_ 

~ 


El 


14  MOMENTS  AND  STRESSES  IN  SLABS. 

In  the  general  case  the  bending  moments  M       and  M    ,  and  the  tors 
moment  if     are  all  present,  and  the  resultant  deformations  are  then 


(5) 


SxSy      El  (7) 

Equations  (5),  (6),  and  (7)  express  the  deformations  in  terms  of  the 
moments.  By  solving  them  with  respect  to  the  moments,  the  moments  are 
found  in  terms  of  the  deformations: 

M  El       (     SZZ          rsS?Z\ 

M*=Mf*(&?-Xfyz)   '  (S) 

.,     El  (  fftfz.    S*z\ 
ty-JTPW&f-Jf*)  .  (9) 

.,       El  (  S?Z\ 
and      Mz=M;\{toSy)  (10) 

By  substituting  these  values  of  the  moments  in  equation  (4),  which  is 
a  relation  between  the  moments  and  the  applied  load,  a  direct  relation  is 
found  between  the  applied  load  «*  and  the  deformations,  .;.  The  result  is 
Lagrange's  equation  for  the  flexure  of  plates,  or,  the  "plate  equation," 


We  may  introduce  Laplace's  operator 

=  Jif    +   6^ 

ox       6y 

which  gives 

^      (Sx4       &r<5yz     Sy4 
Then  Lagrange's  equation    (S),  may  be  written  in  the  simpler  form 


•  (12) 

One  may  determine,  in  a  similar  manner,  a  direct  relation  between  the 
shear  Y  and  the  deflections  zs  by  combination  of  the  equations  (2),  (5), 
(tf),and  (7).  One  finds: 

El    ^AZ 


r.        El 

Yy-~l-K*    Sy     '  ,14) 

Calculations  are  sometimes  made  under  the  assumption  that  Poison's 
ratio  is  equal  to  zero.  Let  Mx<  J/yi  M^  rx,  T'y,  and  c,  denote  the 
moments,  shears,  and  deflections  when  Poisson's  ratio  is  equal  to  zero,  while 
M'x  .  .  .  .  ,  V\  .  .  .  .  ,  ~'  •  •  are  tne  corresponding  values,  for  the  same  load, 
when  Poisson's  ratio  has  a  value  K  which  is  different  from  zero.  There  are 


MOMENTS  AND  STRESSES  IN  SLABS.  15 

certain  relations  between  the  two  sets  of  values  which  apply  when  the 
boundary  of  the  area  under  consideration,  as  marked  by  the  supports  and 
by  the  edges  of  the  plate,  is  fixed,  or  consists  of  simply  supported  straight 
edges,  or  consists  of  parts  which  are  fixed  and  parts  which  are  simply 
supported  along  straight  edges.  These  relations  apply  to  the  slabs  dealt 
with  in  this  report,  but  they  do  not  apply,  in  general,  when  the  supports 
are  elastic,  or  when  there  are  unsupported  edges,  or  simply  supported 
curved  edges.  The  relations  may  be  verified  by  inspection  of  equations  (11) 
or  (12),  (8)  and  (9),  (10),  and  (13)  and  (  14)  ,  respectively.  The  relations 
are: 


(16) 


K/=K  , 


Since  Poisson's  ratio  K  varies  according  to  the  material  used,  it  is 
expedient  to  make  the  calculations  of  £,  j^j       M  ......  ,  etc.,  on  the  basis  of 

K  =  0.    The  values  «',  M1      M'  ......  ,  etc.,  for  any  particular  value  of  K 

may  then  be  determined  afterward  by  formulas  (15  to  (18). 

When  K  =  Q,  then  the  equations  (12),  (5)  to  (10),  (1.3)  and  (14) 
assume  the  simplified  forma: 


S*z\      ,,    „/ S*z  f 

(20) 


(21) 


where    Az  =TTT  -t  -5—? 

o"       oy    '  OA^        oA~oy~     oy ' 

Equations  (19)  to  (22),  in  connection  with  equations  (15)  to  (IS), 
constitute  a  set  of  fundamental  relations,  by  which  plates  may  be  analyzed. 
The  most  difficult  part  of  the  problem  lies  in  the  solution  of  Lagrange's 
equation,  (19).  This  equation  must  be  solved  in  each  case  witli  due  con- 
sideration of  the  particular  boundary  conditions. 

The  reactions  in  a  beam  are  expressed  by  the  end  shears,  and  end 
moments.  In  a  similar  way,  the  reactions  in  a  slab  may  be  expressed  in 
terms  of  the  shears,  bending  moments,  and  torsional  moments  at  the  edge. 
The  torsional  moments  along  a  straight  simply  supported  outer  edge  may 
be  replaced  by  an  equivalent  vertical  reaction  by  the  method  indicated  by 
Kelvin  and  Tait.  The  method  is  described  fully  in  Xadai's  work  on  rec- 
tangular  plates  ( see  the  historical  summary ) . 


16  MOMENTS  AND  STRESSES  IN  SLABS. 

The  above  differential  equations  in  rectangular  coordinates  have  fur- 
nished the  larger  number  of  the  results  which  are  indicated  in  the  following 
articles.  Polar  coordinates  may  be  introduced  instead  of  rectangular 
coordinates  by  transformation  of  the  above  equations;  they  have  been  used 
with  advantage  in  the  analysis  of  circular  slabs  (see  for  example  Foppl's 
treatment).  Beside  the  method  of  differential  equations  two  other  methods 
stand  out  as  effective  in  analysis,  and  they  have  furnished  some  of  the 
results  which  are  quoted  and  used  in  the  following  articles.  These  methods 
are:  Ritz's  method,  which  is  based  on  the  energy  variations  for  the  whole 
plate;  and  the  method  of  difference  equations,  which  was  used  by  Nielsen 
(see  Art.  4) . 

7.  MOMENTS  IN  RECTANGULAR  PLATES  SUPPORTED  ON  FOUR  SIDES. 

The  following  notation  is  used  : 
a  =  longer  span. 

b          =  shorter  span. 

a         =  b/a  =  ratio  of  shorter  span  to  longer  span. 
w         =  uniformly  distributed  load  per  unit-area. 
A/,,^     —  positive  moment  per  unit-width  at  the  center  of  the  panel,  in  the 

direction  of  the  short  span.    J/bc    is  referred  to  as  the  "positive 

moment  in  the  short  span." 
M        —  maximum  positive  moment  per  unit  width  in  the  direction  of  the 

long  span,  or  "maximum  positive  moment  in  the  long  span." 

This  maximum  moment  occurs  somewhere  on  the  center  line 

parallel  to  the  long  sides,  but  not  necessarily  at  the  center  of  the 

panel  (see  the  small  diagrams  at  the  top  in  Fig.  3). 
Mbe     =  negative  moment  per  unit-width  at  the  center  of  the  long  edge,  in 

the  direction  of  the  short  span,  or  "negative  moment  in  the  short 

span." 
Jl/       —  negative  moment  per  unit-widtli  at  the  center  of  the  short  edge, 

in  the  direction  of  the  long  span,  or  "negative  moment  in  the 

long  span." 
M d,      =  moment  per  unit-width  at  the  corner  across  a  line  through  the 

corner,  making   angles   of  45   degrees   with   the   sides    (see   the 

sketches  at  the  top  of  Fig.  3 ) . 

Fig.  3  to  Fig.  11  show  results  of  analyses  of  rectangular  slabs  sup- 
ported on  four  sides.  The  slabs  are  single  panels.  The  edges  are  assumed 
to  remain  undeflected  in  their  original  plane.  The  edges  are  either  simply 
supported  or  fixed,  as  indicated  in  titles  of  the  figures.  The  load  is  uni- 
formly distributed. 

In  Fig.  3  to  Fig.  10  the  abscissas  represent  the  ratio,  oc  ,  of  the 
shorter  span  6  to  the  longer  span  a.  The  right-hand  edge  of  each  diagram 
corresponds  to  cc  =  1,  that  is,  to  a  square  slab,  while  the  left-hand  edge 
corresponds  to  cc  =  0,  or  a  =  oo ,  that  is,  to  an  infinitely  long  slab 
supported  along  the  two  parallel  edges.  The  ordinates  in  Fig.  3  to  Fig.  10 
are  coefficients,  M/irlr,  of  moment  per  unit  width.  The  diagrams  (a),  to 


MOMENTS  AND  STRESSES  IN  SLABS. 


17 


the  left  in  Fig.  3  to  Fig.  8,  show  moments  coefficients  calculated 
by  analysis,  while  the  diagrams  (b),  to  the  right,  consist  of  simplified 
curves  of  approximately  the  same  shape  as  the  curves  to  the  left.  The 
values  indicated  in  the  diagrams  to  the  left  in  Fig.  3  to  Fig.  8,  are  based  on 
a  Poisson's  ratio,  K,  equal  to  zero.  The  points  marked  by  small  squares 
and  triangles  are  based  on  results  found  by  Nadai  and  by  Hencky,  respect- 
ively.* The  points  marked  by  circles  were  determined  in  the  present 
investigation  by  independent  calculations.  In  these  calculations  infinite 
series  were  used  which  are  based  on  Navier's  and  Levy's  solutions!  of 
Lagrange's  equation  ((11),  (12),  or  (19),  in  Art.  6).  The  series  are 
similar,  but  not  identical,  to  those  used  by  Nadai  and  Hencky.  Each  coeffi- 


FIG. 


0  0.2  0.4  0.6  0.8  1.0 

Ratio  of  Short  Span  to  LonqSpan  b/a-a 

(al 
3. — BENDING   MOMENTS   PER   UNIT   WIDTH   IN   RECTANGULAR   SLABS 


o         o.z        0.4        0.6        0.8        ia 
Ratio  of  Short  Span  fo  LonqSpan  Va-oc 


WITH  SIMPLY  SUPPORTED  EDGES. 

Poisson's  ratio  equal  to  zero;    (a)  calculated  values;     (b)   simplified  curves. 

cient  indicated  in  Fig.  3  to  Fig.  8  is  the  outcome  of  a  rather  large  amount  of 
numerical  work.  The  resiilts  shown  in  Fig.  3  to  Fig.  8  might  be  supple- 
mented by  coefficients  which  have  been  determined  by  Leitz,  Mesnager,  and 
lSTielsen,t  some  of  whose  results  will  be  quoted  later.  On  the  whole,  the 
results  obtained  in  the  different  investigations  are  very  consistent. 

The  results  stated  by  Nadai  and  Hencky  are  based  on  a  Poisson's  ratio 
K  =  0.3,  while  the  values  given  in  Fig.  3  to  Fig.  8  are  for  Poisson's  ratio 
equal  to  zero.  Coefficients  which  apply  when  Poisson's  ratio  is  zero  may  be 
derived  by  formulas  (16)  in  Art.  6,  from  the  corresponding  coefficients 
which  apply  when  Poisson's  ratio  has  some  other  value.  Nadai's  and 


*  See  Art.  4,  footnotes  36  and  33,  respectively. 

t  See  Art.  4,  footnotes  5   and   13;    also,  A.   E.  H.   Love,   Mathematical   theory  of 
elasticity,  1906,  p.  468. 

}  See  Art.  4,  footnotes  35,  37,  and  27,  respectively. 


18 


MOMENTS  AND  STRESSES  IN  SLABS. 


llencky's  results  were  transformed  in  this  manner,  as  will  be  shown  by  ait 
example.  Take  the  moments  at  the  center  of  a  simply  supported  slab  with 
6/a  =  O.G.  Nadai,  in  hia  Table  6,  p.  38,  indicates  the  values  M'  =0.1289 
to  (a/2)2  for  the  short  span,  and  M'^  =  0.0704w(a/2)2,  for  the  long  span. 
Formulas  (16),  in  Art.  6,  then  determine  the  corresponding  values  for 
K  —  0 : 


Mv- 


M'    -  KM' 


0.1289-  0.3X0.0704 


wb* 


1  -  K*  I  —  0.32  0.62.22 

and,  in  the  same  way,  J/x  —  0.0243u'62  These  values  of  the  momenta  at 
the  center,  for  cc  =  0.6,  are  indicated  in  Fig.  3  (a).  The  coefficients  given 
in  Fig.  3  to  Fig.  8  are  for  Poisson's  ratio  equal  to  zero. 




±s» 

\ 

s*h 

I 

z 

*h' 

\ 

•    H-0£cct+6cc* 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

-A 

H 

<'& 

ft 

QJa 

r) 

\ 

_ 

-^ 

10 


06 


0.2  0.4  06 

of  5AJ/7*  5oon  to  Long  Span 


Ratio  of  Short  Span  to  Lonq  Span  Va-  a: 

FIG.  4. — BENDING  MOMENTS  PER  UNIT  WIDTH  IN  SIMPLE  SPAN  OF  REC- 
TANGULAR SLAU.S;  Two  PARALLEL  EDGES  FIXED  AND  Two  EDGES 
SIMPLY  SUPPORTED. 

Poisson's  ratio  equal   to  zero;     (a)   calculated  values;     (b)   simplified  curves. 

Fig.  3  deals  with  rectangular  slabs  witli  simply  supported  edges. 
When  a  =  oo  ,  that  is,  cc  =  0,  the  slab  acts  as  a  beam,  and  the  moment 
coefficient  for  the  span  b  becomes  one-eighth.  When  <x  increases  from  zero 
to  one,  then  the  moment  coefficient  A/bc  /wb2  decreases  from  0.1250  to 
0.0369,  and  the  corresponding  coefficient  for  the  center  of  the  long  span 
increases  from  0  to  0.0369.  The  value  0.0369  applies  to  the  square  slab 
(6  =  a,  a  :=:!).  For  this  case  Hencky's  analysis  gives  the  coefficient 
0.0365,  Leitz's  analysis  of  1914  gives  0.0368,  Mesnager's  0.0368,  and  Niel- 
sen's 0.0366.*  In  Fig.  3 (a)  there  is  a  separate  curve  for  the  maximum 


•  See    the   investigations   quoted    in    Art.    4    in    footnotes    33    (Hencky     p 
(Leitz,  p.   27);    37    (Mesnager,  p.  369);    and  27   (Nielsen,  p.   132). 


34);     35 


MOMENTS  AND  STRESSES  IN  SLABS. 


moment  in  the  long  span.  This  curve  defines  coefficients  which  are  greater 
than  the  corresponding  values  at  the  center  of  the  long  span;  that  is,  the 
maximum  moment  in  the  long  span  of  a  rectangular  panel  does  not  occur 
at  the  center,  except  when  the  slab  is  nearly  square.  Moment  diagrams  for 
the  center  line  of  the  long  span  are  shown  in  Fig.  11.  Two  cases  are 
represented,  namely,  oc  =  i£  and  oc  =  0.  The  coefficients  0.0257  and 
0.0234,  which  are  indicated  in  Fig.  3 (a),  appear  as  maximum  ordinates  in 
Fig.  11;  and  the  coefficient  0.0174,  for  oc  =  y2,  appears  in  both  figures,  as 
applying  to  the  moment  at  the  center,  in  the  direction  of  the  long  span. 

A  simply  supported  square  or  rectangular  slab  (simply  supported  on 
four  sides),  when  loaded,  has  a  tendency  to  bend  up  at  the  corners.  In 
the  slabs  treated  here  the  corners  are  assumed  to  be  anchored,  that  is,  the 
supports  provide  for  a  concentrated  downward  reaction  at  each  corner. 
With  this  force  acting,  stresses  and  moments  are  set  up  at  each  corner: 
there  is  a  positive  moment,  M  across  the  line  which  makes  angles  of  45 

deg.  with  the  sides,  that  is,  across  the  diagonal  in  the  square  slab;    there  is 


0  O.Z  04  06  08  1.0 

Ratio  of  Short  Span  to  Long  Span  b/a-a: 

ft 


0  02  0.4  06 

Ratio  of  Short  Span  to  Long  Span  %-fc 

(bl 


FIG.  5. — POSITIVK  BENDING  MOMENTS  PER  UNIT  WIDTH  IN  FIXED  SPAN  OF 
RECTANGULAR  SLAB;  Two  PARALLEL  EDGES  FIXED  AND  Two  EDGES 
SIMPLY  SUPPORTED. 

Poisson's  ratio  equal  to  zero;     (a)  calculated  values;     (b)   simplified  curves. 

an  equally  large  negative  moment  in  the  direction  of  this  line,  and  there  is 
an  equally  large  torsional  moment  in  the  sections  parallel  to  the  sides. 
The  presence  of  negative  moments  in  the  direction  of  the  diagonal  of  a 
square  slab  may  be  understood  easily  when  one  considers  the  curve  of 
deflections  along  the  whole  diagonal.  This  curve  has  a  horizontal  tangent 
at  the  corner,  because  the  deflected  surface  lias  a  horizontal  tangential 
plane  at  this  point.  The  convex  side  of  this  elastic  curve,  therefore,  is 
upward;  that  is,  the  moment  is  negative.  The  following  values  of  the 
coefficients,  MAi  /ivb2,  in  a  square  slab  were  determined :  by  Nadai's 
analysis,  0.0479;  by  Mesnager  (his  paper,  p.  369),  0.0464;  and  by  the 
present  investigation,  0.0463.  The  concentrated  downward  reaction  is  equal 
to  twice  the  diagonal  or  torsional  moment  per  unit  width;  that  is,  the 
present  analysis  leads  to  a  corner  reaction  equal  to  2  X  0.0463wj62  = 
0.0926w&2.  Leitz  indicates  this  reaction  as  0.092w62. 

The  average  coefficient  of  moment  across  the  diagonal  in  a  simply  sup- 


20 


MOMENTS  AND  STRESSES  IN  SLABS. 


ported  square  slab  may  be  determined  by  simple  statical  principles.*  It  is 
1/24  —  0.0417;  that  is,  practically  the  average  of  the  extreme  values, 
0.0463  and  0.0369,  occurring  at  the  corner  and  at  the  center,  respectively. 
The  coefficient  1/24  has  been  used  frequently  as  a  basis  of  design.  This 
value,  1/24,  may  be  justified  on  the  ground  that  when  the  proportional 
limit  is  exceeded,  or  when  the  material  begins  to  yield,  in  a  part  of  the 
diagonal  section,  the  stresses  will  be  redistributed  so  that  they  become  more 
nearly  uniformly  distributed. 

The  curves  in  Fig.  3 (a)  do  not  have  equations  which  can  be  expressed 
by  simple  algebraic  formulas.     It  is  possible,  however,  to  indicate  simple 


^ 

-M* 

jnb* 

"~  I  +0.30:" 

^      \ 

— 

-L 

ng  _ 

ipar 

— 

± 

*4» 

A 

J^ 

s 

She 

rfS 

oan 

l+O.Za:-' 

\ 

*—  —  - 

--. 

s 

. 

y* 

pi* 

b 

\ 

a 

0  02  0.4  0.6  0£ 

Ratio  of  5horf  Span  fo  Longman  tfa-  ex 


0  OZ  04  0£  OS  1.0 

Ratio  ofShorf  Span  to  Long  Span  ¥a~a: 

la)  .Ib) 

FIG.  6. — NEGATIVE  BENDING  MOMENTS  PEE  UNIT  WIDTH  IN  FIXED  SPAN  OF 
KECTANGTJLAB  SLABS;  Two  PARALLEL  EDGES  FIXED  AND  Two  EDGES 
SIMPLY  SUPPOBTED. 

Poisson's  ratio  equal  to  zero;    (a)  calculated  values;    (b)  simplified  curves. 

formulas  which  give  nearly  the  same  values  as  are  found  in  Fig.  3 (a). 
Such  formulas,  and  the  curves  which  represent  them  graphically,  are  indi- 
cated in  Fig.  3(b).  The  formulas  and  the  curves  give  the  coefficient  I/24 
for  the  square  slab.  The  curve  for  Afdla  l'e9  lower  in  Fig.  3(b)  than  in 
Fig.  3 (a).  The  decrease  for  oc  —  0  from  0.0678  in  Fig.  3 (a)  to  0.0625  in 
Fig.  3(b)  may  be  defended  on  the  ground  that  in  very  long  slabs  the 
stresses  at  the  corner,  due  to  Mdla  are  local  stresses  upon  which  the 
safety  of  the  slab  as  a  whole  does  not  depend.  And  in  the  case  of  oc  =  1 
the  probable  redistribution  of  moments  and  stresses  across  the  diagonal, 
when  the  material  begins  to  yield  at  one  point,  will  account  for  the  proposed 
reduction  of  the  coefficient,  from  0.0463  to  1/24.  In  selecting  the  formulas 


"Used  by  Bach  in  his  plate  theory,  see  the  paper  quoted  in  Art.  4,  footnote  15. 


MOMENTS  AND  STRESSES  IN  SLABS. 


21 


indicated  in  Fig.  3(b)  some  weight  was  given  to  the  desirability  of  having 
simple  formulas,  upon  which  design  computations  might  be  based. 

Fig.  4,  Fig.  5,  and  Fig.  6  deal  with  rectangular  slabs  which  have  two 
fixed  opposite  edges  and  two  simply  supported  opposite  edges.  In  a  uni- 
formly loaded  single  continuous  row  of  simply  supported  panels  each  panel 
acts,  on  account  of  the  continuity,  in  the  same  way  as  the  single  panel 
with  two  fixed  and  two  simply  supported  edges.  The  torsional  moments 
and  bending  moments  are  zero  at  the  corners  in  these  slabs.  As  in  Fig. 
3 (a),  separate  curves  are  indicated  in  Fig.  4 (a)  and  Fig.  5 (a)  for  the 
maximum  moments  in  the  long  span;  these  curves  lie,  in  part,  above  the 
corresponding  curves  for  the  moment  at  the  center.  Certain  individual 
points,  which  were  derived  from  Nadai's  work  (p.  62,  Table  9,  in  his  work), 
lie  at  a  distance  from  the  curves  drawn  through  the  rest  of  the  points. 
There  are  three  such  points  lying  below  the  bottom  curve  in  Fig.  4 (a),  and 


.06 


—  .04 


0  02  0.4  0.6  0.8 

Ratio  of  Short  Span  to  Long  Span 


0  02          0.4          0.6          0.8  1.0 

Ratio  of  Short  Span  to  Long  Span  %-a: 


FIG.  7. — POSITIVE  BENDING  MOMENTS  PER  UNIT  WIDTH  IN  RECTANGULAR 
SLABS  WITH  FIXED  EDGES. 

Poisson's  ratio  equal  to  zero;    (a)  calculated  values;     (b)   simplified  curves. 

three  points,  belonging  to  the  same  cases,  lying  above  the  top  curve  in 
Fig.  5  (a).  In  Fig.  6  (a)  one  point  lies  below  the  bottom  curve.  There  is  a 
possibility  of  an  error  in  these  points.  Fig.  6  (a)  shows  the  peculiar  result 
that  greater  negative  moments  are  produced  when  the  long  span  is  fixed 
than  when  the  short  span  is  fixed.  The  simplified  curves  to  the  right  in 
Fig.  4,  Fig.  5  and  Fig.  6  follow  rather  closely  the  curves  to  the  left. 

Fig.  7  and  Fig.  8  deal  with  slabs  fixed  on  four  sides.  Unfortunately, 
this  case,  on  account  of  the  greater  difficulties  involved,  has  been  treated 
less  extensively  than  the  preceding  cases.  Navier's  and  Levy's  solutions 
do  not  apply  to  these  slabs.  Ritz's  method,  which  was  applied  to  these 
slabs,  for  example,  by  Nadai,  leads  to  a  fairly  satisfactory  analysis.  The 
curves  in  Fig.  7  (a)  are  drawn  according  to  Hencky's  results.  For  the 
moments  at  the  center  of  a  square  plate  various  writers  have  indicated 
values,  which  lead  to  the  following  coefficients:  Heneky,  0.0177;  Nadai, 
0.0177;  Mesnager,  0.018;  Leitz,  0.01S4;  Nielsen,  0.0171;  the  present 
investigation,  by  an  approximate  method,  0.0194.  For  the,  negative  moments 
at  the  center  of  the  edge  of  a  square  panel  the  same  writers  have  indicated 


22 


MOMENTS  AND  STRESSES  IN  SLABS. 


the  following  coefficients:*  Hencky,  — 0.0513;  Nadai,  — 0.0487;  Mes- 
nager,  — 0.0474;  Leitz,  —  0.0515;  Nielsen,  — 0.0511;  the  present  investiga- 
tion, by  an  approximate  method,  —  0.0493.  The  curve  for  M  ,  in  Fig. 
7(b),  and  the  line  for  M  in  Fig.  8(b),  have  been  drawn  according  to  an 
estimate,  and  they  may  have  to  be  revised  later. 

Fig.  9  contains  a  summary  of  all  the  simplified  cur^s  in  Fig.  3  to  Fig. 
8.  The  curves  for  the  negative  moments  are  shown  to  the  left,  those  for 
the  positive  moments  to  the  right.  Table  II  gives  a  summary  of  the  for- 
mulas represented  by  these  curves.  A  corresponding  set  of  formulas, 
applying  exactly  to  an  elliptic  plate  with  fixed  edges  and  with  Poisson's 
ratio  equal  to  zero,f  is  indicated,  for  the  purpose  of  comparison  with  the 
other  formulas,  in  the  bottom  line  in  the  table. 


2fc 


M^'-^wb^ 


0  0?  0.4          0.6          0.3  10 

Ratio  ofShorf  Span  -fa  LongSpan  tya-cc 

FIG.  8. — NEGATIVE  BENDING  MOMENTS  PER  UNIT  WIDTH  IN  RECTANGULAR 
SLABS  WITH  FIXED  EDGES. 

Poisson's  ratio  equal  to  zero;    (a)  calculated  values;    (b)  simplified  curves. 

Fig.  10 (a)  illustrates  the  influence  of  change  in  Poisson's  ratio.  Such 
a  change  causes  a  redistribution  of  the  moments  in  the  slab.  The  case 
dealt  with  is  again  that  of  the  slab  with  simply  supported  edges.  Two  of 
the  curves  in  Fig.  3 (a)  are  reproduced,  namely,  the  curve  for  the  moment 
in  the  short  span  at  the  center,  and  the  curve  for  the  moment  at  the  corner, 
in  a  section  making  angles  of  45  degrees  with  the  sides;  that  is,  the  curves 

for   M^     and   M*.       respectively'.     These  curves  are  marked  X  =  0;  those 
DC  alag 

for  K  =  0.3  are  indicated  in  the  figure.  The  computations  of  the  changed 
moment  coefficients  were  made  according  to  forrmilas  (16)  in  Art.  6.  They 
apply  when  Poisson's  ratio,  K,  is  equal  to  zero.  The  corresponding  curves 


*  See  the  investigations  quoted  in  Art.  4  in  footnotes  33  (Hencky,  p.  53) ;  36 
(Nadai,  p.  86);  37  (Mesnager,  p.  413);  35  (Leitz);  27  (Nielsen,  p.  139).  Leitz's 
paper  of  1917,  unfortunately  was  not  available  to  the  writer.  Leitz's  results  for  the 
square  slab  are  quoted  from  Nielsen. 

t  See  A.  Foppl,  Technische  Mechanik,  Vol.  5,  ed.  1918,  p.  106. 


MOMENTS  AND  STRESSES  IN  SLABS. 


23 


change  from  A  =  0  to  K  =  0.3  is  seen  to  increase  the  moment  at  the  center, 
and  to  decrease  the  moment  at  the  corner.  In  the  square  slab  the  moments 
across  the  diagonal  are  redistributed;  the  point  of  maximum  moment 
across  the  diagonal  is  moved  from  the  corner  to  the  center. 

The  stresses  at  a  point  in  a  homogeneous  slab  are  directly  proportional 
to  the  moments  at  the  point.  But  the  maximum  stress  at  a  point  does  not 
necessarily  define  the  "nearness  of  rupture"  or  "tendency  to  failure"  at  the 
point.  This  tendency  depends  on  the  whole  "state  of  stress"  at  the  point, 
or,  in  the  slab,  on  the  state  of  moments  at  the  particular  point.  In  a  square 
slab  the  moments  at  the  center  are  equal  in  all  directions,  while  at  the 


— 

—  - 

—  . 

--- 

«Jfe 

Ik 

^v 

& 

% 

h 

s 

Othe 

^Hfe^ 

**•.., 

L 

v    — 

""•--», 

3 

i 

<3 

^ 

^ 

Other  Span  hxed,  M^ 

-\ 

— 

1      1      1      1 
Full  'Unei-  'Moments  fy  in  theshaltryan 
Dotted  Lin&Mmnh  Ma  in  ttx  longer  yon 
w  -  uniformly  distributed  load, 
a  -  lonq  span, 
b  -short  span 

*:*  

:: 

o         o*        0.4        0.6        o.a        1.0 
Ratio  of  Short  Span  toLonqSpan  b/a-cz 


0  O.Z  0.4  0.6  08  1.0 

Ratio  of  Short  Span  to  Long  Span  h/a~<x. 


P'IG. 


9. — SUMMARY  OF  APPROXIMATE   CURVES  IN  FIG.   3  TO  FIG.   8;     (a) 
NEGATIVE  MOMENTS;    (b)   POSITIVE  MOMENTS. 


corner  a  positive  moment  across  the  diagonal  is  combined  with  an  equally 
large  negative  moment  along  the  diagonal.  Though  the  stresses  may  be 
numerically  larger  at  the  center  than  at  the  corner,  failure,  nevertheless, 
may  be  nearer  at  the  corner,  because  here  the  positive  and  negative  moments 
are  combined.  Various  theories  concerning  the  tendency  to  failure  have 
been  advanced.  One  is  represented  in  Fig.  10 (b);  it  is  the  "shear  and 
strain"  theory,  which  was  originated  by  A.  J.  Becker,*  and  which  was  indi- 
cated by  him  as  applying  to  steel.  Corresponding  to  a  state  of  stress  one 
may  compute  an  "equivalent  stress,"f  which  is  a  simple  tension,  in  one 
direction  only,  which  is  as  dangerous  as  the  given  compound  state  of  stress. 

*  A.    T.    Becker.    The    Strength    and    Stiffness    of    Steel    Under    Biaxial    Loading. 
University  of  Illinois   Eng.   Exp.  Sta.   Bull.  85,  1916. 

+  See,  for  example,  Journal  of  the  Franklin  Institute,  v.   189,   1920,  p.   635. 


24 


MOMENTS  AND  STRESSES  IN  SLABS. 


If  the  shear  and  strain  theory  applies,  the  equivalent  stress  at  a  given  point 
may  be  computed,  as  the  larger  of  the  following  two  quantities:  one  is  the 
modulus  of  elasticity  times  the  greatest  unit-elongation  or  unit-shortening 
at  the  point  in  any  direction;  the  other  is  the  greatest  shearing  stress  at 
the  point,  divided  by  a  certain  constant,  which,  according  to  Becker's 
results,  is  0.6.  An  equivalent  moment  in  a  slab  is  a  bending  moment  which 
would  produce  the  equivalent  stress.  According  to  Becker's  results,  the 
equivalent  moment  at  a  point  is  computed,  then,  as  the  larger  of  the  follow- 
ing two  quantities:  one  is  El  times  the  numerically  largest  curvature  in 
any  vertical  section  at  the  point;  the  other  is  the  largest  torsional  moment 


Hff*  0656) 
Sending  Moments  at 
Comer  across  ttie 
Diagonal 


0  02  04  06  03  10  0  02  04  06  OS 

Ratio  of  Short  Span  to  Long  Span  ^/a-tz  Ratio  of  Short  Span  to  Long  Span1: 

(a)  (b) 

FIG.   10 (a). — INFLUENCE  OF  VARIATION  IN  POISSON'S  RATIO,  K,  ON  THE 

MOMENTS  IN  RECTANGULAR  SLABS  WITH  SIMPLY  SUPPORTED  EDGES. 

FIG.  10 (b). — "EQUIVALENT  MOMENTS,"  BASED  ON  THE  "SHEAR  AND  STRAIN" 

THEORY,  IN  RECTANGULAR  SLABS  WITH  SIMPLY  SUPPORTED  EDGES. 


in  ai^y  section  at  the  point,  divided  by  0.6.  Such  equivalent  moments  are 
indicated  in  Fig.  10 (b).  The  slabs  are  the  same  as  in  Fig.  10 (a),  and  the 
curves  refer  to  the  center  and  to  the  corner.  The  methods  of  computation 
are  indicated  in  the  figure.  According  to  equations  (15)  and  (20)  in  Art. 
6,  El  times  the  curvature  may  be  computed  as  ( 1  —  K-)  times  the  moment 
corresponding  to  Poisson's  ratio  equal  to  zero.  M  and  M  „  ,  as  used  in 
(he  notes  in  Fig.  10(b),  are  the  moments  corresponding  to  Poisson's  ratio 
equal  to  zero.  The  curves  in  Fig.  10 (b)  explain  why  a  square  slab  with 
K  =  0.3  may  fail  at  the  corners  first,  in  spite  of  the  fact  that  the  stresses 
are  smaller  at  the  corner  than  at  the  center. 

Fig.  10 fa)   and  Fig.  10 (b)   and  the  discussion  in  connection  with  these 
figures  Bhow  how  the  results  derived  for  the  case  in  which  Poisson's  ratio  is 


MOMENTS  AND  STRESSES  IN  SLABS. 


25 


zero  may  be  interpreted  and  used  when  Poisson's  ratio  has  any  other  value, 
provided  the  law  of  failure  of  the  material  is  known.  Whether  or  not  the 
curves  and  formulas  indicated  in  Fig.  3  to  Fig.  9,  in  Fig.  11,  and  in  Table 
II  may  be  applied  as  a  basis  for  design  of  actual  slabs,  should  be  deter- 
mined for  the  individual  materials  by  comparison  with  experimental 
results. 

8.  MOMENTS  IN  SQUARE  INTERIOR  PANELS  OF  UNIFORMLY  LOADED  FLAT 
SLABS. 

Notation : 

I  =  span,  measured  from  center  to  center  of  the  columns. 

c  =  diameter  of  the  column  capitals. 

w  =  load  per  unit-area,  uniformly  distributed  over  all  panels. 
W  =  total  panel  load. 

The  slab  under  consideration  is  a  girderless  or  "flat"  slab,  supported 
directly  on  the  column  capitals,  which  are  assumed  to  be  round.  Lines 
connecting  the  centers  of  the  columns  divide  the  floor  into  square  panels, 


=3E3? 


o    .2b 


6b   .8b    b 


tzb 


l.4b  I6b 


idb  sb 


Distance  a/ong  x- 

@ 


FIG.  11. — MOMENTS  ALONG  THE  CENTER  LINE  OF  THE  LONG  SPAN  IN  REC- 
TANGULAR SLABS   WITH   SIMPLY  SUPPORTED   EDGES. 


with  a  column  at  each  point  of  intersection.  An  interior  panel  is  consid- 
ered. It  is  surrounded  on  all  sides  by  similar  panels,  all  carrying  the  same 
load.  For  the  convenience  of  the  analysis  it  may  be  assumed  that  there  is 
an  infinite  number  of  equal,  square  panels,  all  carrying  the  same  load.  The 
slab  is  assumed  to  be  fixed  in  the  column  capitals  at  the  edge  of  each  column 
capital.  A  panel  of  this  description,  loaded  as  indicated  here,  will  be 
referred  to  as  "a  normal  panel."  On  account  of  the  symmetry,  the  column 
capitals  supporting  the  normal  panel  will  not  tend  to  rotate  about  a  hori- 
zontal axis,  the  tangents  across  the  edges  and  center  lines  of  the  normal 
panel  will  remain  horizontal,  and  the  torsional  moments  along  the  center 
lines  of  the  panel  and  along  the  parts  of  the  edges  between  the  column 
capitals  will  be  zero. 

Fig.  12  shows  sections  for  which  it  lias  become  customary  to  indicate 
the  moments.  The  terms  column-head  sections,  mid-section,  outer  sections. 
and  inner  section  are  in  accordance  with  common  practice.  The  moments 
in  these  sections  are  taken  in  a  direction  perpendicular  to  the  straight  parts 
pf  the  sections.  Moment  coefficients  for  tbese  sections  will  be  stated  pres- 


26  MOMENTS  AND  STRESSES  IN  SLABS. 

ently,  but  first  some  remarks  will  be  made  concerning  the  processes  of  the 
analysis. 

The  present  investigation  is  built,  in  part,  on  certain  results  which 
were  found  by  N.  J.  Nielsen  *  in  his  analysis  of  plates  by  the  method  of 
difference  equations.  Nielsen  analyzed  various  types  of  square  interior 
panels  of  uniformly  loaded  flat  slabs:  first,  point-supported  slabs,  in  which 
tli£  column  capitals  and  the  columns  have  been  reduced  to  point  supports; 
second,  slabs  in  which  the  supporting  forces  are  uniformly  distributed  within 
squares  with  side  0.21  and  with  the  centers  at  the  centers  of  the  column; 
third  and  fourth,  slabs  supported  on  square  column  capitals  with  the  sides 
0.21  and  OAl,  respectively;  fifth,  a  slab  with  dropped  panels  (areas  of  in- 
creased thickness  around  the  supports)  ;  and  sixth,  a  slab  in  which  there  is  no 
bending  resistance  across  the  central  parts  of  the  edges  of  the  panels,  that 
is,  no  bending  resistance  in  parts  of  the  mid-sections.  Nielsen  divided  the 
panel  into  elementary  squares  with  side  \  =  0.11.  l/6i,  0.21,  or  0.251.  The 

Column-head    M/d-Secr/on  CaJu/nn  -head 
Secf/on) \Secf/on 


]  Outer  '      Inner      i  Outer 
\5ecfiffr    Section    S0:f/on\ 


FIG.  12.—  MOMENT  SECTIONS  FOR  FLAT-SLAB  PANELS. 

variables  in  the  equations  are  the  deflections  at  the  corners  of  these  squares; 
one  equation  is  indicated  for  each  such  corner.  Since  finite  squares  are  used 
instead  of  infinitesimal  rectangular  elements  the  method  is  approximate, 
not  exact,  as  applying  to  homogeneous  slabs.f  The  smaller  the  value  of  \ 
the  closer  is  the  approximation.  The  value  \  =  O.ll  gives,  on  the  whole, 
rather  satisfactory  results.  \  =  1/61  to  0.251  gives  results,  use  of  which 
may  be  made  in  comparative  studies  of  distributions  of  deflections  and 
moments  in  different  slabs;  such  use  of  some  of  Nielsen's  results  will  be 
made  later  (in  Art.  10).  But  the  moment  coefficients  found  with  \  = 
1/61  or  more,  hardly  seem  to  be  sufficiently  exact  when  considered  as  inde- 
pendent results  applying  to  fundamental  cases.  For  this  reason  the  use  of 
Nielsen's  results  was  limited  here  to  those  obtained  with  \  =  O.ll.  This 
value  of  X  was  used  by  Nielsen  only  in  the  first  two  of  the  cases  men- 


*  N.  J.  Nielsen,  Bestemmebe  af  Spaendinger  i  Plader  ved  Anvendelse  af  Differen- 
sligninger,  1920  (referred  to  in  Art.  4,  footnote  27). 

t  The  difference  equations  apply  exactly  to  a  certain  rib-structure  in  which  the 
bending  deformations  are  concentrated  at  the  points  of  intersection  of  the  ribs,  and 
in  which  the  torsional  resistance  is  supplied  by  special  structural  elements  which  con- 
nect one  rib  with  another. 


MOMENTS  AND  STRESSES  IN  SLABS. 


27 


tioned,  that  is,  in  the  analyses  of  the  point-supported  slab  and  of  the  slab 
with  the  supporting  forces  uniformly  distributed  within  small  squares; 
in  the  other  cases  he  used  \  >  O.H.  Those  of  Nielsen's  results,  with 
X  r=  O.H,  which  apply  to  the  point-supported  slab,  were  represented  graph- 

TABLE   IT. — APPROXIMATE   FORMULAS   FOR   BENDING   MOMENTS   PER   UNIT 

WIDTH  IN  RECTANGULAR  SLABS  AND  ELLIPTIC  SLABS 

SUPPORTED  ON  THE  PERIPHERY. 


The  Tormulas  are  represented  qrophicatlLimfiq31orT<i9. 
a  -/onqerspan,  b  -  shatter  span ;ot- Obi  Fbissons  rot/o-O. 


Momenta  in  span  b. 


/It  center '  /?/"  center 
of  edqe.      of  5/ab. 


Moments  in  span  g 


At  center 

of  edqe 

~Mae 


A/onq  center 
fine  of  slob. 


Fburedqes 

simply 

supported. 


wb 


1+ZoC3 


Span  & 
fried; 

-Span  g_ 
simple. 


1-t-OAoC? 


o 


wb^ 
60 


Span  g_ 

f/xed; 

Spanb. 

simple. 


ItO-Boc 


All 
edc/es  fixed. 


£llipt/c  j  ~/ab 
wiih  fixededqe; 
diameters 


j> 


f+jof+a* 


-0C 


ically  for  the  purpose  of  the  present  investigation,  and  adjustments  were 
made,  similar  to  those  by  which  a  string  polygon  is  modified  into  a  string 
curve.  By  such  adjustments  of  the  curves  for  moments  and  deflections  it 
was  possible  to  improve  the  approximation  slightly.  It  would  be  possible 
to  analyze  the  point-supported  slab  by  differential  equations,  and  to  obtain, 
thereby,  an  increased  degree  of  exactness,  but  in  this  study  it  seemed 
desirable  to  make  use  of  the  available  results.  The  degree  of  approximation 
obtained  by  this  use  may  be  judged  by  comparing  the  moment  coefficients 
for  square  slabs  supported  on  four  sides,  found  by  Xielsen  with  \  equal  to 
one-tenth  of  the  side,  as  quoted  in  the  preceding  article,  with  the  corre- 
sponding moment  coefficients  found  in  other  analyses. 


28 


MOMENTS  AND  STRESSES  IN  SLABS. 


The  use  of  Nielsen's  results  in  connection  with  results  found  in  the 
present  investigation  by  means  of  the  differential  equations  will  now  be 
described.  The  point-supported  slab  is  not  important  in  itself,  because 
actual  slabs  do  not  have  point  supports.  But  the  results  found  for  the 
point-supported  slab  may  be  used  in  the  analysis  of  the  normal  panel,  sup- 
ported on  round  column  capitals,  in  the  same  way  as  moment  diagrams  for 
simple  beams  are  used  in  the  study  of  continuous  beams  or  beams  with 
fixed  ends.  The  diagrams  for  fixed  beams  can  be  found  from  the  diagrams 
for  simple  beams  by  adding  the  effects  of  the  end  moments.  The  simple 
beam  is  considered  in  this  connection  as  a  "substitute  structure"  which 
temporarily  replaces  the  given  fixed  beam,  and  which  is  made  to  act  like  the 
given  beam  by  adding  the  end  moments.  In  a  similar  way  the  point- 
supported  slab  may  be  used  as  a  substitute  structure  which  temporarily 
replaces  the  slab  supported  on  column  capitals,  and  which  is  made  to  act 
like  the  original  slab,  that  is,  have  the  same  deflections  and  moments  at  all 


J 

2 

3 

v 

I 

T 

^ 

4- 

_7 

m& 

£- 

I 

T 

7\ 


FIG.  13. — MOMENT  SECTIONS  REFERRED  TO  IN  TABLE  IV;    THE  HEAVY  SHORT 
LINES  INDICATE  THE  POSITIONS  AND  DIRECTIONS  OF  THE  SECTIONS. 


points  outside  the  circles  marked  by  the  edges  of  the  column  capitals,  by 
adding  certain  loads  near  the  center  of  the  column.  The  nature  and 
intensity  of  these  loads  must  be  such  that  the  resultant  deflections  and 
slopes  at  the  circles  marking  the  edges  of  the  column  capitals  become  zero. 
The  essential  part  of  the  load  of  this  kind,  added  at  each  column,  may  be 
termed  a  "ring  load."  It  consists  of  a  combination  of  an  upward,  force 
uniformly  distributed  over  the  circumference  of  a  small  circle,  drawn  on 
the  slab,  with  an  equally  large  downward  force  at  the  center  of  this  circle. 
The  ring  load  may  be  considered  as  concentrated,  just  as  a  couple  acting  on 
a  beam  may  bo  considered  as  concentrated  at  one  point,  for  example,  at  the 
end  point  if  the  end  is  fixed.  A  ring  load  of  the  proper  intensity  at  each 
point  support,  combined  with  a  certain  uniformly  distributed  bending 
moment  applied  at  the  edge  of  the  whole  slab  (which  includes  many  panel?) 
and,  combined  with  the  original  uniformly  distributed  load  and  the  reac- 


MOMENTS  AND  STRESSES  IN  SLABS. 


29 


TABLE  III. — PERCENTAGES  OF  SUM  OF  POSITIVE  AND  NEGATIVE  MOMENTS 
RESISTED  IN  SECTIONS  SHOWN  IN  FIG.  12. 

Results  of  analysis  of  a  square  interior  panel  of  a  uniformly  loaded  homogeneous  flat  slab. 
The  sum  of  positive  and  negative  moments  is  approximately  equal  to 


c=diameter  of  column  capital;  /=span;   W=  total  panel  load;  Poisson's  ratio  =0. 


c/l 

Average. 

0.15 

0.20 

0.25 

0.30 

Negative  Moments 
Positive  Moments 

Column  head  section 

Across  edge  of  capita! 
Outside  capital  

31.8 
16.5 
48.3 
17.0 
65.3 
20.9 
13.8 
34.7 

37.1 
11.3 
48.4 
16.7 
65.1 
20.9 
14.0 
34.9 

40.2 
S.I 
48.3 
16.6 
64.9 
20.8 
14.3 
35.1 

42.7 
5.7 
48.4 
16.3 
64.7 
20.7 
14.6 
35.3 

20+80  c/l 
28—80  c/l 
48 
17 
65 
21 
14 
35 

Total  .  ... 

Total  negative  mome 
Outer  section 

it  

Inner  section  

Total  positive  momes 

t  

TABLE  IV.  —  COEFFICIENTS  OF  MOMENT  PER  UNIT  WIDTH,  IN- 
SERTIONS SHOWN  IN  FIG.  13. 

Results  of  analysis  of  a  square  interior  panel  of  a  uniformly  loaded  homogeneous  flat  slab.    Poisson's 
ratio  =0. 

The  coefficients  are  values  of  the  expression 

M 


They  are  found  by  multiplying  the  coefficients  in  Fig.  14  to  16  by  the  factors  8,  9.52,  10.31,  11  .  39,  and 
12.46  for  c/J  =  0,  0.15,  0.20,  0.25,  and  0.30,  respectively. 


e/l- 

Approximate 
Coefficients 

Section. 

0 

0.15 

0.20 

0.25 

0.30 

forc/J  = 
0.15  to  0.30 

1  .     ..     .          

—2.120 

—1  854 

—  1  609 

—  1.424 

-H  C-J-+0 

2  

—0  488 

—0  461 

—0.435 

—0.409 

—0.370 

^  c      ' 
—0.5—  1.5(  -fV 

3  

—0  255 

—0.270 

—0.275 

—0.282 

—0.289 

\l  1 

—0.028 

4 

0  258 

0  130 

0  048 

—0  046 

—0.169 

0.23—  4.5(4  V 

5... 

0  015 

0  015 

0  015 

0.016 

0.015 

V  i  J 

+0.02 

6... 

0  474 

0  474 

0.466 

0.461 

0.447 

0.46 

7  

0.331 

0.341 

0.340 

0.343 

0.343 

0.34 

8. 

0  226 

0  242 

0  246 

0.254 

0  262 

0.25 

9  

0.233 

0  188 

0.156 

0.121 

0.072 

0.23—  l.s(4Y 

10  

—0.121 

—0.110 

—0.102 

—0.093 

—0.080 

\lj 
K-K(|V 

\l  / 

30  MOMENTS  AND  STRESSES  IN  SLABS. 

lions  at  the  point  supports,  will  make  the  slab  deflect  in  such  a  way  that 
the  circles  marking  the  edges  of  the  column  capitals  practically  become  a 
contour  line  along  which  the  tangential  planes  are  horizontal  and  coin- 
ciding. By  introducing  certain  supplementary  loads,  beside  the  ring  loads, 
the  conditions  of  the  edges  of  the  column  capitals  may  be  satisfied  with  any 
desired  degree  of  approximation.  Corrections  by  means  of  such  supple- 
mentary loads  were  omitted,  because  the  degree  of  approximation  obtained 
without  these  loads  appeared  to  be  acceptable;  besides,  since  the  degree  of 
approximation  is  limited  in  one  part  of  the  problem  by  the  use  of  the 
approximate  results  found  by  difference  equations,  the  gain  by  a  further 
increase  of  exactness  in  the  part  of  the  problem  discussed  here  would  be 
only  slight. 

It  remained^  then,  to  investigate  the  effects  of  the  ring  loads,  to  deter- 
mine their  intensity,  and  to  make  the  proper  additions  to  the  moments  in 
the  point-supported  slab.  The  method  used  was  that  of  differential  equa- 
tions. Lagrange's  equation  ((11),  (12),  or  (19),  in  Art.  6)  was  solved 
for  the  case  of  the  ring  loads  by  double-infinite  series,  and  the  moments  at 
definite  points,  produced  by  the  ring  loads,  were  computed  by  corresponding 
double-infinite  series.  As  in  the  preceding  article,  and  for  similar  reasons, 
Poisson's  ratio  was  taken  as  zero  (compare,  in  particular,  the  discussion 
made  in  connection  with  Fig.  10  (a)  ).  Details  of  this  analysis  will  be  pre- 
sented in  Appendix  A. 

The  results  will  now  be  described.  Reference  is  made  again  to  Fig.  12, 
which  shows  the  customary  moment  sections.  Table  III  gives  results 
found  for  these  sections.  The  moments  are  stated  in  per  cent  of  the  sum 
of  the  numerical  values  of  positive  and  negative  moments  in  all  the  sections 
in  Fig.  12.  Values  are  given  for  four  sizes  of  the  column  capital.  The  per- 
centages resisted  in  the  different  sections  in  Fig.  12  are  seen  to  change 
only  slightly  when  c  changes  from  0.151  to  0.30Z,  and  to  deviate  only  slightly 
from  the  constant  values  given  in  the  last  column :  namely,  48  per  cent  in 
the  column-head  sections,  17  per  cent  in  the  mid-section,  21  per  cent  in  the 
outer  sections,  and  14  per  cent  in  the  inner  section. 

The  total  moments  in  the  various  sections  depend  upon  the  moments 
per  unit-width  at  the  individual  points  of  the  slab.  Fig.  13  indicates 
points  and  sections  at  which  the  moments  per  unit-width  are  of  particular 
interest.  Moment  coefficients  for  these  sections  are  stated  in  Table  IV. 

Fig.  14  shows  diagrams  of  coefficients  of  moment  per  unit-width  across 
the  edge  and  the  center  line  of  the  panel.  The  coefficients  are  values  of 
Jf/wP. 

Lavoinne,*  in  a  paper  published  in  1872,  derived  the  stresses  in  certain 
sections  of  a  uniformly  loaded  point-supported  slab.  He  stated  coefficients 
of  stresses;  but  moment  coefficients  MAc/l-,  of  the  type  used  in  Fig.  14, 
may  be  found  by  dividing  his  stress  coefficients  by  six.  Thus,  Lavoinne's 


*  Lavoinne,   Sur  la   resistance  des  paroi?  planes  des  chaudieres  a   vapenr,   Annal 
Fonts  et  Chaussees,   1872,   pp.   276-295;    the  numerical  coefficients  are  quoted   frc 
?86.     See  the  historical  summary  in  Art.  4,  footnote   10. 


MOMENTS  AND  STRESSES  IN  SLABS. 


31 


analysis  gives  the  following  moment  coefficients:  at  the  center  of  the  slab, 
0.17/6  =  0.0283,  to  be  compared  with  0.0283  in  Fig.  14;  at  the  center  of 
the  edge  along  the  edge,  0.34/6  —  0.0567,  while  Fig.  14  gives  0.0592;  at 
the  center  of  the  edge  across  the  edge,  —  0.20/6  =  —0.0333,  while  Fig.  14 
gives  — 0.0319.  Since  Lavoinne  stated  only  two  decimal  places  in  each 


FlG.  14.— COEFFICIENTS  OF  BENDING  MOMENTS  PER  UNIT  WIDTH  IN  A 
SQUARE  INTERIOR  PANEL  OF  A  UNIFORMLY  LOADED  FLAT  SLAB  WHEN 
POISSON'S  RATIO  is  ZERO;  MOMENTS  ACROSS  THE  EDGE  AND  THE 
CENTER  LINE. 


coefficient,    the    agreement    may    be    considered     as    fairly     satisfactory. 
Lavoinne's  solution  is  by  double-infinite  trigonometric  series. 

Fig.  15  shows  coefficients,  M/wl~,  of  moment  per  unit-width  along 
the  edge  and  the  center  line  of  the  panel.  Eacli  of  the  curves  must  satisfy 
a  certain  condition,  which  applies  also  to  beams  with  fixed  ends:  the 
positive  and  the  negative  area  of  each  of  the  moment  diagrams  must  be 


32 


MOMENTS  AND  STRESSES  IN  SLABS. 


Fi«.  15. — COEFFICIENTS  OF  BENDING  MOMENTS  PER  UNIT  WIDTH  IN  A 
SQUARE  INTERIOR  PANEL  OF  A  UNIFORMLY  LOADED  FLAT  SLAB  WHEN 
POISSON'S  RATIO  is  ZERO;  MOMENTS  ALONG  THE  EDGE  AND  THE  CENTEB 
LINE. 


MOMENTS  AND  'STRESSES  IN  SLABS. 


33 


numerically  equal.  This  condition  applies  to  the  curves  in  Fig.  15  because 
the  slope  of  the  slab  is  zero  at  the  edge  of  the  column  capital  and  at 
the  center,  that  is,  at  the  points  where  the  curves  in  the  diagram  end;  the 
application  of  the  condition  under  these  circumstances  follows  from  equa- 


FIG.  16. — COEFFICIENTS  OF  BENDING  MOMENTS  PEE  UNIT  WIDTH  IN  A 
SQUABE  INTERIOR  PANEL  OF  A  UNIFORMLY  LOADED  FLAT  SLAB  WHEN 
POISSON'S  RATIO  is  ZERO;  MOMENTS  ACROSS  AND  ALONG  THE 
DIAGONALS. 

tions    (20)    in  Art.  6.     An  examination  of  the  curves  in   Fig.   15   showed 
equality  of  the  positive  and  negative  areas. 

Fig.  16  shows  coefficients  of  moment  per  unit-width  acrosa  and  along 
the  diagonals.  In  drawing  the  ciirves  for  the  momenta  along  the  diagonal, 
use  was  made  of  the  condition  that  the  positive  and  negative  parts  of  each 


34 


MOMENTS  AND  STRESSES  IN  SLABS. 


moment  diagram  must  be  numerically  equal.  The  diagonal  momenta  at  the 
edge  of  the  column  capital  could  not  be  determined  with  the  degree  of 
exactness  obtained  elsewhere,  because  the  torsional  moments  at  these  points, 
in  the  point  supported  slab,  in  sections  parallel  to  the  panel  edges,  had  not 
been  determined  by  the  difference  equations  with  the  same  degree  of  exact- 
ness as  other  moments.  The  negative  moment  across  the  edge  of  the  column 
capital  is  approximately  the  same  along  the  diagonal,  as  along  the  panel 
edge,  in  fact,  it  is  approximately  constant  all  the  way  around  the  column 
capital.  Accordingly,  the  coefficients  of  end-moment  along  the  diagonal, 
stated  in  Fig.  16,  were  taken  as  equal  to  the  corresponding  negative 
moment  coefficients  in  Fig.  15,  which  apply  at  the  panel  edge.  Certain 
small  discrepancies,  which  possibly  may  be  explained  by  the  deviations  in 
the  negative  diagonal  moments,  will  be  discussed  in  connection  with  the 
next  two  figures. 

Mi. 


OZnl3 


Moments 

FIG.  17. — DIAGRAM  SHOWING  THE  EQUILIBRIUM  OF  THE  FORCES  AND  COUPLES 
ACTING  ON  AN  OCTANT  AND  A  QUADRANT  OF  A  SQUARE  PANEL 
(c  =  0.30). 


Fig.  17  shows  forces  and  couples  acting  on  two  separate  octants  of  a 
normal  panel  with  c  =  0.30Z.  For  each  octant  the  downward  resultants  of 
the  applied  loads  acting  at  the  centroids  of  the  area,  and  the  upward  resultant 
of  the  vertical  supporting  forces  or  shears  at  the  edge  of  the  column  capital 

form  a  couple;    these  couples  are:  M    ,       for  one  octant,  M'    ,       for 

i  -fn  i  +11 

the  other.  In  the  plane  vertical  sections  there  are  bending  moments,  jV/m 
Mlv,  and  Myi  in  one  octant,  M'lir  M'1V,  and  A/'VI  i"  the  other,  but 
on  account  of  the  symmetry  there  are  no  torsional  moments  and  no  vertical 
shears  in  these  sections.  The  moment  Afy  and  M'  are  the  resultant 
moments  in  the  curved  sections  at  the  edge  of  the  column  capital.  In  the 
diagram  to  the  right,  in  the  figure,  the  different  couples  are  represented  as 
vectors.  Each  vector  is  laid  off  parallel  to  the  vertical  plane  of  the  couple 
which  it  represents;  the  direction  is  that  of  the  upper  one  of  two  horizontal 
force*  representing  the  couple.  The  couple  vectors  M  and  M',.,, 


MOMENTS  AND  STRESSES  IN  SLABS. 


are  shown  resolved  into  the  components  i/r  and  MU,  M\  and  M'n. 
The  couple  vectors  form  two  polygons,  one  for  each  octant,  but  with  the 
side  Jfyi  in  common.  Eacli  octant  is  in  equilibrium;  therefore,  if  the 
couples  are  represented  correctly,  each  polygon  should  close.  Fig.  17  shows 
small  gaps  at  the  end  points  of  the  vectors  M  and  M>  These  gaps 

are  a  measure  of  the  discrepancies  which  have  entered,  so  far,  into  the 
calculations  and  into  the  particular  graphical  representation  in  the  figure. 
The  method  of  obtaining  the  particular  vectors  represented  in  the 
vector  diagram  in  Fig.  17  will  now  be  described,  and  possible  sources  of  the 
gaps  will  be  discussed.  The  couple  MI,  with  the  components  M  and 
Afjj,  is  considered  first.  The  two  components  could  be  computed  exactly 
if  the  point  of  application  of  the  resultant  shear  were  known  exactly. 
Afj  and  Mu  were  computed  under  the  assumption  that  the  vertical  shear 
at  the  edge  of  the  column  capital  is  uniformly  distributed,  or,  that  the 
resultant  passes  -through  the  centroid  of  the  circular  arc  formed  by  the 


Afr 


c^ 


C-QZOl 


'.'j 


Scale 


0 '  01  .0?Wl 

FIG.  18. — COUPLES  ACTING  ON  AN  OCTANT  OF  A  SQUARE  PANEL  (c  =  0.15; 

0.20;    0.25). 

section.  The  distribution  of  the  shear  at  the  edge  of  the  column  capital 
is  known  to  be  approximately  uniform,  but  if  it  is  not  entirely  uniform, 
the  end  point  of  M.  may  have  to  be  moved  slightly.  It  is  possible  that  a 
shifting  of  the  end  of  M  into  its  correct  position  would  reduce  the  gap 

were  determined  by 


at  the  end  of  M, 


The  vectors  M        and 


measuring  areas  in  Fig.  14,  of  the  diagrams  of  moments  across  the  center 
line  and  the  edge.  Afy  was  determined  by  a  corresponding  area  in  Fig. 
16.  M  was  computed  under  the  assumption  that  the  bending  moment 
across  the  edge  of  the  column-  capital  is  constant,  and  that  the  torsional 
moment  along  the  edge  of  the  column  capital  is  zero.  If  these  assumptions 
are  correct,  the  direction  of  lfy  will  bisect  the  45  deg.  angle  between  the 
panel  edge  and  the  diagonal.  But  the  assumption  of  even  distribution  is 
not  more  than  approximately  correct;  as  stated  before,  the  diagonal 
moment  across  the  edge  is  not  known  with  the  degree  of  exactness  obtained 
elsewhere.  By  a  slight  change  in  the  bending  moments  and  by  introducing 
small  torsional  moments,  If  may  be  changed  so  as  to  eliminate  the  gap 
between  M  and  M  In  fact,  the  gap  may  be  eliminated,  practically, 

by  changing  the  direction  of  the  couple  jl/y  slightly  without  changing  the 
magnitude. 


36  MOMENTS  AND  STRESSES  IN  SLABS. 

Fig.  18  shows  vector  polygons  of  the  same  kind  as  those  shown  in  the 
preceding  figures.  The  diagrams  apply  to  slabs  with  c/l  =  0.15,  0.20,  and 
0.25.  As  in  the  preceding  figure,  gaps  are  left  open  between  M  and  Al 

The  method  used  here  in  the  study  of  the  equilibrium  of  the  resultant 
couples  acting  upon  an  octant  of  the  slab  is  analogous  to  that  used  by 
J.  R.  Nichols  *  in  his  study  of  the  moments  in  a  quadrant  of  the  slab. 
In  fact,  the  analysis  represented  in  Fig.  17  and  Fig.  18  may  be  looked  upon 
as  Nichols's  analysis  applied  to  an  octant  of  the  slab.  The  sum  of  positive 
and  negative  moments  indicated  in  Fig.  17  is  the  total  moment  indicated 
by  Nichols's  analysis.  The  approximate  value  which  Nichols  gave  in  the 
discussion  of  his  paper  t 

M0  =  \Wl(l  -  \?-)*  (23) 

o  O    ( 

is  so  nearly  equal  to  the  value  which  he  derived  originally  that  it  may  be 
used  instead;  it  is  the  value  used  in  connection  with  Table  3  and  stated 
at  the  head  of  this  table. 

The  moment  coefficients  for  the  column-head  sections,  mid-section,  outer 
sections,  and  inner  section,  as  taken  directly  from  the  diagrams  in  Fig.  14, 
Fig.  15,  and  Fig.  16,  led  to  the  gaps  in  the  polygons  in  Fig.  17  and  Fig.  18. 
While  a  part  of  the  discrepancy  may  be  due  to  a  slight  error  in  the  total 
moment,  and  while  it  is  possible  that  the  main  part  of  the  discrepancy  is 
due  to  the  unevenness  in  the  distribution  of  moments  at  the  edge  of  the 
column  capital,  it  \vas  considered  feasible  to  make  adjustments  by  correct- 
ing each  bending  moment  in  proportion  to  its  size.  That  is,  the  percen- 
tages of  total  moment  indicated  in  Table  3  were  left  unchanged;  they  are 
the  original  percentages  based  on  areas  measured  in  the  diagrams  in  Fig. 
14.  Adjustments  of  the  coefficients  of  moment  per  unit-width,  stated  in 
Table  4,  were  introduced  by  a  correction  of  the  factors  which  are  stated 
at  the  head  of  the  table,  and  which  were  used  in  transforming  the  coeffi- 
cients M /wV,  of  the  type  used  in  the  diagrams,  into  coefficients  of  the  type 
used  in  Table  IV. 

9.  UNBALANCED  LOADS  ON  FLAT  SLABS.  The  load  on  one  panel  of  a 
flat-slab  floor-structure  has  some  influence  on  the  stresses  in  the  adjoining 
panels.  If  a  load  which  is  originally  uniformly  distributed  over  all  panels 
is  changed  by  removing  or  reducing  the  loads  on  some  of  the  panels,  the 
stresses  in  the  remaining  panels  will  be  increased  in  some  sections  and 
decreased  in  others.  The  loads,  by  this  change,  become  unbalanced. 
Unbalanced  loads  cause  the  tangents  across  the  panel  edges  to  rotate;  they 
may  produce  bending  moments  in  the  columns  or  may  cause  the  column 
capitals  to  rotate  about  horizontal  axes.  Unbalanced  loads  on  continuous 
beams  produce  analogous  effects;  for  example,  the  positive  moments  in  one 
span  increase  when  the  downward  loads  in  the  two  adjacent  spans  are 
removed.  In  the  analysis  of  flat-slab  structures  the  varying  degree  of 


•  See  Art.  4,  footnote  17. 

t  J.  R.  Nichols,  Discussion  on  reinforced-concrete  flat-slab  floors.     Am.  Soc.  C.  E., 
v.  77,  1914,  p.  1735. 


MOMENTS  AND  STRESSES  IN  SLABS. 


37 


stiffness  of  the  columns  must  be  taken  into  consideration;    this  stiffness  of 
the  columns  affects  the  stresses  under  unbalanced  loads. 

Fig.  19  and  Fig.  20  show  certain  results  of  the  study  of  unbalanced 
loads.  Further  studies  of  the  effects  of  these  loads  are  made  in  connection 
with  Fig.  21  to  Fig.  24. 


Ratio  of  Loads 

FIG.    19. — BENDING   MOMENTS  IN   OUTER   SECTIONS   DUE   TO   UNBALANCED 
LOADS;    DIFFERENT  UNIFORM  LOADS  ON  ALTERNATE  Hows  OF  PANELS. 


A  flat-slab  structure  is  considered  in  which  each  floor  consists  of  a 
large  number  of  equal  square  panels.  For  the  sake  of  convenience  of 
analysis  the  number  of  panels  may  be  assumed  to  be  infinite  in  all  direc- 
tions, as  in  the  preceding  article.  All  the  columns  are  assumed  to  be  alike. 
Poisson's  ratio  is  again  assumed  to  be  equal  to  zero.  The  small  figures  at 
the  bottoms  of  Fig.  19  and  Fig.  20  show  the  loading  arrangement  on  one 
floor:  each  alternate  row  of  panels  carries  the  full  uniform  load,  w  per 
unit-area,  the  other  rows  carry  the  reduced  uniform  load,  w  per  unit-area. 


38 


MOMENTS  AND  STRESSES  IN  SLABS. 


The  positive  moments  in  the  outer  and  inner  sections  shown  in  Fig.  19  and 
Fig.  20  reach  extreme  conditions  when  w  is  as  large  as  possible,  and  when 
w0  is  as  small  as  possible,  for  example,  when  WQ  Is  equal  to  the  dead  load 
only.  The  abscissas  in  Fig.  19  and  Fig.  20  represent  the  ratio  w  /to  of  the 
The  left-hand  edges  correspond  to  w  =  0,  that  is,  every  other  row 

50 


loads. 


A  .6 

Ratio  of  Load s  ^/ 
FIG.   20. — BENDING   MOMENTS   IN   INNER   SECTIONS   DUE   TO   UNBALANCED 
LOADS;    DIFFERENT  UNIFORM  LOADS  ON  ALTERNATE  Rows  OF  PANELS. 

of  panels  is  entirely  unloaded.  The  right-hand  edges  correspond  to  w  •=.  to, 
that  is,  uniform  load  tr  on  all  panels  as  in  the  preceding  article.  The 
ordinates  in  Fig.  19  and  Fig.  20  represent  percentages  of  the  total  moment 

1  2 

MO  =  -  Wl  (1 c)2;    M0  is  the    sum    of    the  numerical  values  of    the 

8  o 

moments  in  the  outer  sections,  inner  section,  column-head  sections,  and  mid- 
section  when  the  panel  is  loaded  by  w.  The  values  indicated  on  the  right- 


MOMENTS  AND  STRESSES  IN  SLABS. 


39 


hand  edges,  21.0  in  Fig.  19,  and  14.0  in  Fig.  20,  are  the  percentages  apply- 
ing to  the  condition  of  uniform  load  over  all  panels.  These  two  per- 
centages are  approximate  values,  taken  from  the  last  column  in  Table  III 
in  the  preceding  article;  they  are  nearly  equal  to  the  exact  values,  which 
vary  only  slightly  within  the  range  of  variation  of  c/l.  The  two  upper 
pencils  of  lines  in  Fig.  19  refer  to  the  fully  loaded  panels,  the  panels  loaded 
by  w;  the  two  lower  pencils  refer  to  the  panels  which  carry  only  the  par- 
tial load  wo.  Two  extreme  cases  are  represented,  one  by  the  two  middle 
pencils,  the  other  by  the  two  outer  pencils.  In  one  extreme  case  the  col- 
umns are  perfectly  rigid,  and  in  the  other  the  column  capitals  are  perfectly 
free  to  rotate  about  horizontal  lines.  The  latter  condition  may  be  estab- 
lished by  introducing  hinges  in  the  columns  directly  below  the  column 
capitals  and  directly  above  the  slab.  Actual  slab-structures  fall  between 
Hie  two  extreme  conditions,  which,  therefore,  are  analyzed  first.  Methods 
by  which  one  may  interpolate  between  the  extreme  cases  will  be  indicated 
later. 


(a)  (b) 

FIG.   21. — UNBALANCED   LOADS   PRODUCING   MAXIMUM   POSITIVE   MOMENTS 

IN  THE  SLAB; 

(a)  total  applied  load; 

(b)  component,  +  -^,    of  applied  load. 

(c)  component,  ±  — ,   of  applied  load; 

According  to  Fig.  19,  when  w  is  zero,  and  when  the  columns  are  rigid, 
the  percentage  of  moment  in  the  outer  sections  in  the  loaded  panels  ranges 
from  20.3  for  c  =z  OM  to  23.5  for  c  — 0.15L  In  the  unloaded  panels  the 
corresponding  percentage  is  small,  ranging  from  +  0.3  to  —  2.6,  the  latter 
figure  indicating  a  small  negative  moment.  When  the  columns  are  free 
to  turn,  the  load  on  one  panel  has  a  marked  influence  on  the  moments  in 
the  other  panels.  According  to  Fig.  19,  when  w  is  zero,  the  percentage 
for  the  outer  section  ranges  from  41.4  to  49.6  for  the  loaded  panels,  and 
from  —  20.4  to  —  28.6  for  the  unloaded  panels.  The  negative  percentages 
represent  fairly  large  negative  moments.  The  use  of  the  diagram  may  be 
illustrated  by  an  example.  Assume  w  —  QAw,  c  =  0.2l;  then  if  = 


121  2 

-  wl  (I  -  -  c)2  =  -  wl*  (1  -  - .  0.2)2  =   0.0939t0Z3. 


The  diagram  gives 
the  following  values  of  the  moment  in  the  outer  sections:     in  the  fully 


40  MOMENTS  AND  STRESSES  IN  SLABS. 

loaded  panels,  0.218M  when  the  columns  are  rigid,  0.347M  when  the 
column  capitals  are  free  to  turn;  in  the  partially  loaded  panels,  0.075Af0 
when  the  columns  are  rigid,  — 'o.053M0w^en  the  column  capitals  are  free 
to  turn. 

Fig.  20,  which  refers  to  the  inner  section,  is  analogous  to  Fig.  19. 
The  percentage  of  moment  varies  in  the  same  general  manner  as  in  Fig.  19. 

The  procedure  of  the  analysis  will  now  be  discussed.  The  load  consist- 
ing of  w  and  to  on  alternate  rows  of  panels  is  denoted  by  w,w  The 
moments  produced  by  this  load  are  linear  functions  of  w  Q  ;  and  when  w 
ia  considered  as  a  constant,  they  are  linear  functions  of  the  ratio  wo  /w. 

It  follows  that  when  the  moments  for  the  extreme  values  w     =0  and 

o 

w  =w  are  known,  that  is,  the  moments  represented  at  the  left  and  right 
edges  in  Fig.  19  and  Fig.  20  are  known,  then  the  moments  for  intermediate 
values  may  be  determined  by  linear  interpolation,  as  represented  by  the 
straight  lines  in  the  two  figures.  It  remains,  therefore,  to  make  an  analysis 
for  the  load  w,0,  (or  w  and  zero  in  alternate  rows  of  panels) .  This  load,  w,0, 
may  be  resolved  into  two  components  by  the  scheme  indicated  in  Fig.  21 
and  Fig.  23.*  The  load  w,Q  on  each  floor  in  Fig.  21  (a)  is  resolved  into  two 

romponents:     -f- — ,  shown   in   Fig.  21  (b),  uniformly  distributed  over  all 
2 

panels;    and  -|-  —  ——   shown  in  Fig.  21  (c),  consisting  of  the  upward  and 

2*      2' 
downward    uniform    loads   t/;/2    on    alternate    rows    of    panels.      The    load 

_ is  anti-symmetrical  with  respect  to  the  dividing  edges,  that  is,  the 

2'        2 

edges   at  which  the   load   changes  from   -f- w'/2   to  — w/2.     The  structure 

itself   is  symmetrical  with   respect  to  the  vertical   sections  through   these 

edges.     Consequently,  under  the  load  -}-  _ the  deflections  at  points 

2'        2' 

which  are  symmetrical  with  respect  to  the  dividing  edges  are  equal  and 
opposite;  the  dividing  edges  remain  straight  and  undeflected;  the  moments 
in  sections  which  are  symmetrical  with  respect  to  the  dividing  edges  are 
equal  and  opposite;  and  the  moments  at  the  dividing  edges  are  zero. 

When  the  column  capitals  are  free  to  turn,  and  when  their  diameter 

is  gmall,  the  slab,  under  the  load  -f-  —    —  —     will  deflect  within  each  row 

2'        2' 

of  panels  as  if  that  particular  row  were  separated  from  the  rest  of  the 
slab,  and  as  if  it  were  simply  supported  on  girders  at  the  two  parallel 
edges  of  the  row.  The  moment  per  unit-width  across  the  center  line  of 

the  row,  accordingly,  is  -f- I"  in  the  row  loaded  by  -}-  _.    and  — -  —  /' 

89  O  '  Q    O 

W 

in  the  row  loaded  by  —  _      Since  the  column  capitals  are  in  the  regions 

2' 
near  the  points  of  inflection,  a  change  in  the  size  of  the  diameter  of  the 


•  This  scheme  was  used  by  Nielsen   (Spaendinger  i  Plader,  1920,  p.   192). 


MOMENTS  AND  STRESSES  IN  SLABS.  41 

column  capitals,  even  to  the  greatest  size  c  =  0.31,  has  only  a  slight  influ- 
ence on  the  state  of  flexure  of  the  slab  under  the  load  -|-  —  —  —  as  long 

2'        2' 

as  the  column  capitals  are  free  to  turn.  Minor  local  redistributions  of  the 
moments  occur  near  the  edges  of  the  column  capitals  as  a  result  of  this 
change  in  the  diameter  of  the  column  capital,  but  in  the  inner  and  outer 
sections  the  influence  of  this  change  is  negligible.  That  is,  the  values 

.t 12  may  be  used  without  reference  to  the  size  of  the  column  capital. 

8  2 

The  method  of  calculation  for  the  load  w,Q,  when  the  column  capitals  are 
free  to  turn,  may  be  shown  by  an  example.  Take  c  =  0.21.  The  corre- 
sponding total  moment  is  M  =  -  wl3  (1  -  -.  0.2)2  =  0.0939wZ3.  The 

°        8  3 

W  W  1     W  I 

moments  in  the  outer  sections  due  to  the  load  +    — ,  ~  —  are   =*=    -  •  —  •  I2  •  - 

wl3 
=    ±    — —   =  0.333Af0.      The  moment  in  the  outer  sections,   due  to  the 

w    w         1 
uniform  load    — ,   — ,  is  -  .  0.21  Af0  =  0.105A/0.      The  resultant  moments  in 

£t         £  £ 

the  outer  sections  are  then:  in  the  loaded  panels,  0.333A/0  +  0.105A/0  = 
0.438Afo;  in  the  unloaded  panels  _  0.333AT,,  +  0.105A/0  =  _  0  228A/0. 
The  percentages  43.S  and  —  22.8  are  shown  at  the  left-hand  edge  in  Fig.  19. 
The  other  percentages  represented  at  the  left-hand  edges  in  Fig.  19  and 
Fig.  20  were  calculated  in  the  same  manner. 

The  slab-structure  with  rigid  columns  and  immovable  column  capitals 
was  analyzed  as  a  statically  indeterminate  structure  in  which  the  turning 
couples  transferred  from  the  slab  through  the  column  capital  to  the  column 
are  introduced  as  statically  indeterminate  quantities.  The  structure  with 
column  capitals  free  to  turn  is  used  in  the  analysis  as  a  substitute  structure. 
In  this  substitute  structure  the  column  capitals  turn  under  the  influence 

of   the    load    _| _  but    the    slope    of    the    column    capitals    may   be 

reduced  to  zero,  and  thus  the  substitute  structure  may  be  made  to  act  like 
the  original  structure,  by  applying  a  turning  couple  of  the  proper  magni- 
tude and  direction  at  each  column  capital.  The  resultant  moments  in  the 
slab  are  found,  then,  by  adding  the  moments  produced  by  the  turning 
couples  to  those  already  existing.  In  order  to  find  the  magnitude  of  the 
turning  couples  and  the  effects  of  them,  a  study  was  made  of  bending 
moments  and  slopes  produced  by  turning  couples  ±  21  of  constant  magni- 
tude, applied  at  the  column  capitals.  Results  of  this  study  are  stated  in 
Table  V(a).  Tn  order  to  derive  these  results  the  structure  with  column 
capitals  free  to  turn  was  replaced  by  a  second  substitute  structure  in  which 
the  column  capitals  are  removed  altogether.  This  second  substitute  struc- 
ture is  the  same  point-supported  slab  that  was  used  in  the  preceding  article 
in  the  study  of  the  slab  with  the  same  load  in  all  panels.  The  second  sub- 
stitute structure  is  loaded  at  the  points  of  support  by  concentrated  couples 


42 


MOMENTS  AND  STRESSES  IN  SLABS. 


±  21  and  by  certain  additional  concentrated  loads  which  may  be  called  ring 
couples.  A  ring  couple  may  be  obtained  by  applying  two  equal  and  two 
opposite  ring  loads,  of  the  kind  described  in  the  preceding  article,  so  close 
together  that  the  whole  system  of  forces  may  be  considered  as  a  concen- 
trated load.  The  concentrated  couples  acting  alone  do  not  produce  uniform 
slopes  at  the  circles  marking  the  edges  of  the  column  capitals;  but  uni- 
formity of  the  slopes  at  these  circles  is  restored  by  adding  ring  couples  of 
the  proper  intensity.  The  second  substitute  structure  is  thus  made  to  act 
like  the  first  substitute  structure,  which  is  the  slab  with  column  capitals 
which  are  free  to  turn  relative  to  the  columns.  The  influence  of  the  ring 
couples  on  the  moments  at  the  center  line  of  the  row  is  small;  it  is  meas- 
ured by  the  difference  between  the  moments  stated  in  the  first  column  in 


TABLE  V  (a). — PENDING  MOMENTS  AND  f LOPES  DUE  TO  TURNING  COUPLES 
+21  APPLIED  AT  THE  COLUMN  CAPITALS. 

The  turning  couples  are  clockwise  and  counter-clockwise  in  alternate  rows  of  columns. 

i  is  perpendicular  to  the  rows,  y  is  in  the  direction  of  the  rows.    The  couples  are  in  planes  parallel  to  zt. 

*  =  gpan;  c  =  diameter  of  column  capital;   /  =  moment  of  inertia  per  unit-width;  Poisson's  ratio =0. 


ell 

0 

0.15 

0.20 

0.25 

0.30 

Momenta-direction  (  <££  °ff  «*£"**  tO  '"  '  ' 

0.886 
1  081 

0.898 
1  073 

0.907 
1  066 

0.920 
1  057 

0.936 
1  046 

per  unit-width.           |  Average  for  width  1  

1  0 

1  0 

1  0 

1  0 

1  0 

Moments  in  v-direction  {  <££  J  ^"^  *°  *'  '  ' 

0.294 
—  0  248 

0.282 
—  0  240 

0.273 
—  0  233 

0.260 
—  0  224 

0.244 
—  0  213 

per  unit-width.            |  Average  for  length  L  .......     .. 

0 

0 

0. 

0 

0 

0  469; 

0  472* 

0  475? 

0  478* 

0  482* 

0  531* 

0  5281 

0.525.' 

0  522* 

0  518* 

Slope  »  of  column  capital,  in  i-direction  

1.081* 

0.975* 

0.834* 

Approximate    values    of   «,    calculated   by    the   formula 
i  —  c 

2EI 

2E1 

2EI 

'    2-0  82EI                                 • 

1.037* 

0.975* 

0  .  854* 

2EI 

2EI 

2EI 

Table  V(a)  and  the  moments  in  the  remaining  columns.  The  moments  and 
slopes  in  Table  V(a)  were  calculated  by  means  of  infinite  series  which  are 
solutions  of  Lagrange's  differential  equation;  details  of  this  analysis  will 
be  given  in  the  appendix. 

The   slope   of   the   column    capitals   under   the   influence   of   the    load 

-f  -, ,  or  the  load  w,  0,  is  found  to  be,  with    close   approximation, 


wl3 
4&EI 


('"i(f)1)- 


(24) 


The  values  of  the  slope  s  of  the  column  capitals  under  the  influence  of  the 
couples  ±  21  are  stated  in  Table  V(a).  The  turning  couples  ~^  Mc 
which  are  necessary  to  make  the  resultant  slope  of  the  column  capital 


MOMENTS  AND  STRESSES  IN  SLABS.  43 

o 

equal  to  zero  is  determined,  then  by  the  formula  MC  =  2  1.   -  (25) 

8 

M c  is  the  resultant  couple  which  is  transferred  through  each  column  capital, 
from  the  columns  to  the  slab  in  the  original  structure,  which  is  the  struc- 
ture with  rigid  columns  and  immovable  column  capitals.  An  example  will 
show  the  manner  of  computing  Mc  and  the  resultant  bending  momenta 
which  are  shown  at  the  left-hand  edges  in  Fig.  19  and  Fig.  20.  Tako 

c  =  0.2Z.      Equation    (24)    gives  SQ  —   — —     .     0.985.      Table   V(a)    gives 

0  9751  48EI 

s  =    -  —  .   The  couple  transferred  through  each  column  capital  is,  then, 

according  to  formula  (25),  Me=  — -=  ^-  .--  =  0.0842wZ3  =  0.897A/0. 

s          0.97o     \2i 

According  to  Table  V(a)  the  moment  in  the  inner  section  in  the  slab  with 
column  capitals  free  to  turn,  produced  by  the  turning  couples  ^  21  is 
=*=  0.525?.  The  corresponding  moment  produced  by  the  turning  couples 

-  0.525Z  .  Mc          -  1 
+  Mc  is  then  +  -  -   =  +  -    .    0.525  .  0.897 M0  =  +  0.236A/0. 

—  I  J-i 

The  signs,  minus  and  plus,  refer  to  the  loaded  and  unloaded  row  of  panels, 
respectively.  According  to  Fig.  20  the  moments  in  the  inner  section  in  the 
slab  with  column  capitals  free  to  turn,  produced  by  the  load  w,0,  are 
0.403A/  in  the  loaded  row  of  panels  and  —  0.263Af  in  the  unloaded  row. 

O 

The  resultant  moments  in  the  inner  sections  in  the  slab  with  rigid  columns 
are  then: 

in  the  loaded  row  of  panels:  0.403A/0  _  0.236A/0  =  0.167 MQ- 
in  the  unloaded  row  of  panels:    _  0.263MQ  +  0.236A/0    =  -0.027 A/0. 
The  coefficients   0.167   and  — 0.027   are  expressed  as  percentages  and  are 
shown  at  the  left-hand  edge  in  Fig.  20.     The  remaining  percentages  belong- 
ing to  the  two  middle  pencils  in  Fig.  19  and  Fig.  20  were  computed  in  a 
similar  manner. 

Two  extreme  cases  have  been  considered  so  far:  one  with  perfectly 
stiff  columns  and  fixed  column  capitals;  the  other  with  columns  which  are 
flexible  or  supplied  with  hinges  at  the  ends,  so  as  to  allow  the  column 
capitals  to  turn  freely  with  the  slab.  Actual  slab-structures  have  an  inter- 
mediate degree  of  rigidity  of  the  column  capitals.  These  structures  can  be 
dealt  with  if  a  method  of  interpolation  between  the  extreme  cases  can  be 
devised,  so  that  one  can  say  that  a  given  case  belongs,  for  example,  70  per 
cent  or  0.7  to  one  extreme  case,  and  30  per  cent  or  0.3  to  the  other.  For 
the  purpose  of  the  interpolation  two  definite  ratios  are  introduced  measur- 
ing the  degree  of  fixity  of  the  column  capitals  and  the  degree  of  freedom  of 
the  column  capitals  to  rotate.  These  ratios  are 

k  =  fixity  of  the  column  capitals, 

k'  =  1  —  k  =  freedom  of  the  column  capitals  to  rotate. 
These  ratios  are  defined  as  follows:    Let  M  denote  the  moment  in  a  certain 
section,  A/,  and  A/B   the  moments  which  would  occur  in  the  same  section 
if  the  column  capitals  were  fixed  and  free  to  turn,  respectively.     The  ratios 


44 


MOMENTS  AND  STRESSES  IN  SLABS. 


fc  and  k'  are  defined,  then,  as  far  as  the  particular  section  is  concerned,  by 
the  equations: 

M**kMA+k'MBt  (26) 

fc+fc'-l.  (27) 

For  example,  M  =  130000  in.  Ib,  M A  =  100000  in.  Ib,  and  MB  =  200000 
in.  Ib,  gives  k  =  0.7,  k'  =  3 ;  the  column  capitals  may  be  said  to  be  70 
per  cent  fixed  and  30  per  cent  free  to  turn.  M  may  be  calculated  by  formula 
(26)  when  MA,  A/B,  k  and  k'  are  known.  The  limiting  values  of  k  and  k' 
are  0  and  1;  the  combination  k  =  1,  k'  =  0  represents  the  extreme  case  of 
fixed  capitals;  the  combination  k  =  0,  k'  =  1  represents  the  case  of  column 
capitals  which  are  free  to  turn. 

The  values  of  k  and  k'  may  be  different  at  the  different  moment  sec- 


FKJ.   22. — MOMENTS   IN   COLUMNS   DUE  TO  UNBALANCED  LOAD  SHOWN  IN 

FIG.  21; 

(a)   columns   without   capitals; 
(h)   columns  with  capitals; 

tions.  But  in  certain  important,  rases,  for  example,  in  those  shown  in  Fig. 
21  and  Fig.  23,  /.-  and  k'  are  independent  of  the  position  of  the  moment 
section;  k  and  k',  then,  are  constants  belonging  to  this  structure  as  a 
whole.  Let 

M'  ~  moment  transferred  from  a  column  through  the  column  capital 

to  the  slab  in  the  structure  with  intermediate  rigidity  of  the 

column    capitals; 
M  —  moment  transferred   from  a  column  through  the  column  capital 

to  the  slab  in  the  structure  with  fixed  column  capitals. 
Application  of  (2(5)   to  these  moments  gives 

A/'c-We  (28) 

where  fr  is  the  fixity  k  referring  to  the  moments  which  are  transferred 
through  the  column  capitals.  Assume  that  a  calculation  by  (28)  leads  to 


MOMENTS  AND  STRESSES  IN  SLABS. 


45 


the  same  value  of  k  for  all  the  column  capitals  within  an  area  which 
includes  a  large  number  of  panels  in  botli  directions.  It  may  be  shown 
that  A;  in  (26),  under  this  condition,  is  the  same  for  all  sections  and  is 
equal  to  the  constant  value  k  ;  the  moment  M  in  any  section  may  be 
expressed  as  a  linear  function  of  the  moments  M'  ',  by  substituting 
M'  _.  k  M  ,  the  moment  M  becomes  a  linear  function  of  ke',  according 
to  (26)  and.  (27)  M  is  a  linear  function  of  k;  the  limits  of  k  and  kc 
are  the  same,  0  and  1 ;  consequently,  k  =  k  as  was  to  be  proved.  In 
Fig.  21  and  Fig.  23  M'  and  M  are  the  same  in  all  columns,  except  for 
the  direction  clockwise  or  counter-clockwise;  that  is,  in  each  slab,  k  is  the 
same  at  all  column  capitals  and  in  all  moment  sections.  The  loading 
arrangement  in  Fig.  21  produces  the  greatest  possible  moments  in  the  outer 
and  inner  sections;  the  arrangement  in  Fig.  23  produces  large  moments 
in  the  columns. 


1 

L. 7 ,|,  7  -J.  7 I  ^ > 

(a)  (b)  (ci 

FIG.  23. — UNBALANCED  LOADS  PRODUCING  LARGE  MOMENTS  IN  COLUMNS; 

(a)  total  applied  load; 

(b)  component,  + -^,  of  applied  load; 

(c)  component,  ±  — ,    of  applied  load. 

£ 

The  analysis  is  facilitated  by  resolving  the  loads  ivQ  shown   in   Fig. 

21  (a)    and  Fig.  23 (a)    in  each  case  into  two  components:    namely,^-,  — , 

2t     £ 

shown  in  Fig.  21  (b)   and  Fig.  23(b),  and  !fl,  -  —}  shown  in  Fig.  21  (c)  and 

£          £ 

Fig.  23 (c).  To  make  the  analysis  simple,  the  dimensions,  including  the 
diameter  of  the  columns,  are  assumed  to  be  the  same  through  several 
stories. 

The  dimensions  and  the  loads  are  denoted  as  in  Fig.  21  and  Fig.  23. 
Fig.  22  and  Fig.  24  show  diagrams  of  the  bending  moments  in  the  columns; 
they  show  also  the  notation  for  these  bending  moments.  The  moments  of 
inertia  of  the  sections  are 

T  =  moment  of  inertia  of  the  cross  section  of  the  slab  for  the  whole 
panel  width, 

«7  =  moment  of  inertia  of  the  columns. 

A  simplified  case  is  considered  first,  in  which  the  slab-structure  is 
replaced  by  a  frame  whose  girders  and  columns  have  the  same  moments  of 
inertia,  /'  and  J,  as  the  slab  structure;  at  the  joints  there  are  no  column 


46 


MOMENTS  AND  STRESSES  IN  SLABS. 


capitals,  but  the  connections  between  the  four  adjoining  members  are  rigid. 
Tin:  moment  diagrams  of  the  columns  are  shown  in  Fig.  22 (a)  and  Fig. 
24  (u).  Analysis,  for  example,  by  the  method  of  least  work,  by  the  method 
of  tlio  substitute  structure,  or  by  the  slope-deflection  method,  leads  to  the 
following  values: 

In  Case  I,  Fig.  21  and  Fig.  22  (a),  with 


K 


•me  finds 


that  is, 


Jl 

I'h' 


.wl*  d_      1     \ 

-24  I1    I+K) 


k  -  1  - 


1+K  ' 


1+K 


(29) 
(30) 
(31) 


to)  (b) 

Fin.  24.- — MOMENTS  IN  COLUMNS  DUE  TO  UNBALANCED  LOAD,  SHOWN  IN 


FIG.  23; 


(a)  columns  without  capitals; 

(b)  columns  with  capitals 


In  Case  II,  Fig.  23  and  Fig.  24  (a),  again  with 

Jl 


K  - 


one  finds 


that  is, 


24 


I'h" 
1  - 


(32) 
(33) 


MOMENTS  AND  STRESSES  IN  SLABS.  47 

In  the  slab-structure  the  bending  moments  are  not  uniformly  distrib- 
uted over  the  width  of  the  sections.  One  may  say,  that  the  moment  of 
inertia  /'  of  the  section  is  not  fully  effective.  The  presence  of  the  column 
capitals,  on  the  other  hand,  has  the  effect  of  reducing  the  clear  spans  and 
increasing  the  rigidity  of  the  structure.  The  slopes  or  angles  of  rotation 
of  the  column  capitals  in  Fig.  21  and  Fig.  22  (b)  must  be  equal  or  equal 
and  opposite.  The  moment  diagram  of  the  cylindrical  part  of  the  column 
between  the  top  of  the  slab  and  the  bottom  of  the  column  capital  above, 
must  have  its  centroid,  therefore,  at  the  center  of  the  total  distance  meas- 
ured between  the  centers  of  the  slabs.  In  Fig.  2-1  (b)  the  point  of  inflection 
is  at  the  center  of  the  cylindrical  part  of  the  column.  The  dimensions  of 
the  moment  diagrams  in  Fig.  22 (b)  and  Fig.  24 (b)  have  been  computed 
without  considering  the  influence  of  the  thickness  of  the  slab  upon  tin; 
stiffness  of  the  column.  This  influence  may  be  taken  into  account  by 
measuring  h  from  the  top  of  the  slab  to  the  bottom  of  the  column  capital 
above,  i  from  the  bottom  of  the  column  capital  to  the  bottom  of  the  slab; 
7t'=  h  -f-  i  is,  accordingly,  the  clear  distance  from  the  top  of  one  slab  to 
the  bottom  of  the  slab  above. 

The  slope  of  the  column  at  the  capital  must  be  equal  to  the  angle  of 
rotation  or  slope  of  the  capital.  In  terms  of  the  moments  in  the  columns, 
the  slopes,  Q,  of  the  columns  at  the  capitals  are 

0  °  irwr  in  FiS-  22  (b)'     °  -  £wr  in  Fig  24  (b)'  (  34) 

£i  fLJ  O  C<J 

In  terms  of  the  moments  ±  2X,  transferred  from  the  columns  through  the 
capitals  to  the  slab,  and  in  terms  of  the  applied  load  w,0  or  _j_  ^  _  }-.f 

—  ^- 

the  angle  of  rotation,  or  slope,  of  the  column  capitals  in  Fig.  21  to  Fig.  24 
may  be  expressed  approximately  as  follows,  as  may  be  seen  by  comparison 
with  the  last  line  in  Table  V(a)  and  with  formula  (24)  : 

* 


2  -0.80  -1.02E7'     '    48-1.02#/' 

By  equating  Q  to  Q>  and  substituting  the  values  of  X  given  on  the  diagrams 
in  Fig.  22  (b)  and  Fig.  24  (b),  in  terms  of  X',  the  following  values;  are 
obtained : 

In  Case  I,  Fig.  21  and  Fig.  22  (b),  one  finds 

x' ~  24 !~"+  /1+3i\ 2  "A "_~c\"_  :  JT ;;;  (30) 

The  fixity  k  of  the  column  capitals  is  the  ratio  of  this  quantity  to  the.  value 
of  the  same  quantity  when  J  =  oo  that  is, 

k  -1  -  -JL         A/ L_  (S7J 

l+K'  l+K' 


48  MOMENTS  AND  STRESSES  IN  SLABS. 

where 


(38) 


0.80-1.02  I'h 

In  case  II,  Fig.  23  and  Fig.  24  (b),  one  finds 

Jl 
x,      wl3 1.  021' h  ,39) 

"    24    1  j.   (\  +1\    /I  -  £\  JZ 

3  +   \         /i/\         F/       0.80  -1.027'/i 

The  fixity  k  of  the  column  capitals  is  determined  as  in  Case  I  by  comparing 
X'  with  its  value  when  J  =  oo  one  finds 


A-  =  1  —  k'  •=  (40) 

1  +  3K  '  1  +  3K  ' 


where 


j£  _    _\ IL/__  j__     LZ.         "  (41) 

0.80-     1.02  7'/t 

When  the  fixity  has  been  computed,  the  maximum  moment  X'  in  the  cylin- 
drical part  of  the  column  may  be  computed  as 

X'      —  _^?0fc_ 

24 


Sample  calculations  of  freedom  of  the  column  capitals  to  rotate,  fixity 
of  the  capitals,  and  moments  in  the  columns,  according  to  formulas  (26) 
to  (42),  are  shown  in  Table  V(b).  In  examples  (1)  the  columns  are  rather 
slender,  in  examples  (2)  and  (3)  they  are  comparatively  stiff.  The  mate- 
rial is  assumed  to  be  homogeneous.  The  dimensions  i  should  be  measured, 
as  stated  before,  as  the  vertical  distance  between  the  bottom  of  the  column 
capital  and  the  bottom  of  the  slab.  In  example  ( 1 ) ,  Case  I,  the  column 
capitals  are  found  to  be  about  28  per  cent  fixed,  72  per  cent  free  to  turn. 
Fig.  19  and  Fig.  20  show  that  with  these  degrees  of  fixity  and  freedom  to 
rotate,  unbalanced  loads  will  have  a  considerable  influence  on  the  bending 
moments.  In  examples  (3),  Case  I,  the  fixity  is  90  per  cent;  that  is,  the 
moments  in  the  inner  and  outer  sections  of  the  loaded  panels  will  not  differ 
greatly  from  the  moments  under  uniform  load. 

It  should  be  noted  that  those  approximate  values  of  fixity,  freedom  to 
rotate,  and  moments  in  the  column,  which  are  computed  by  considering 
the  structure  as  a  frame,  do  not  differ  greatly  from  the  corresponding 
values  in  the  lower  part  of  Table  V(b),  which  are  calculated  more  exactly. 
One  may  conclude  that  other  flat-slab  structures  may  be  analyzed,  in  most 
cases  with  a  satisfactory  approximation,  as  far  as  the  moments  in  the 
columns  are  concerned,  by  methods  applying  to  frames.  For  example,  the 


MOMENTS  AND  STRESSES  IN  SLABS. 


49 


TABLE  V(b).  —  SAMPLE  CALCULATIONS  OF  THE  RIGIDITY  OF  COLUMN 

CAPITALS. 


T/re   /rjcr/e/-/er/  /s  ersstfnee/  fe  6e   Aosnevf/reot/s.    Tftf 
/octets  ore   /tre//ictrtK/  //j  /-/f.2**/    and  f/a.JtSf^f.   7#f 
of  r/te  co/umns   ane  ' 


moment  of  inert/a  of  s/ot>  />er  /»*ne/  nf/'t///if 
moment  of  /nerf/a   of  co/t/mns; 
fixity  of  coAumrt  c<yo/7a/s; 
'freeeto/rr   o/  co/t/nrn    caprta/s   to 


//O. 


fv 


ofco/umr. 


/  Sfory 
r/jett  o 
0/amfffr   of  co/umn.  </ 


O.ffOl 


s/nucfufv  tv/'/A  c 


6.Z3 


Case  I 


o.r/s 


0/37 
0.4&3 


O  096 
O.904- 


0459 


0.03+ 


ffo/aevt///7  co/um/i  X-A'- 


3.   /7a/--S/ao    structure 
c 

i 
/r 


aosz 


0.31 
O./02 


O.Z2 

0.041 
07/1 


0.32 
0-662 


O.Z2 
0.462 


0.32 

0091 

0-4/1 


Case  I 


K  = 


0.7Z0 
O.Z00 


OL350 


0.7  4/ 


6.ZZ 

0./S4 

0.36Z 


.69 


0.09ft 
0.904 


0O98 
O.902. 


/ff.36 


rtf.  24T6 


0.4S/ 
O.  £"49 
0. 02.17 


0.47/ 


0.049 


O.OZ27 


0.0375' 


0.948 
O.O397 


30./O 
0032 
0.968 
O.O37/ 


0033 
O  967 
O.O373 


50  MOMENTS  AND  STRESSES  IN  SLABS. 

slope-deflection  method  *  may  be  used  to  advantage  when  the  design  is  less 
uniform  than  assumed  in  Table  V(b)  ;  for  example,  when  the  diameter  of 
the  columns  is  different  in  the  different  stories. 

So  far,  attention  has  been  given  mainly  to  the  moments  due  to  unbal- 
anced loads  in  the  outer  and  inner  sections.  The  negative  moments  in  the 
column-head  sections  and  mid-section  are  affected  less  by  unbalanced  loads 
than  are  the  positive  moments.  In  a  frame  structure  the  maximum  nega- 
tive moment  in  the  second-floor  girder  between  the  third  and  fourth  span 
occurs  with  the  following  combination  of  loaded  and  unloaded  spans;  W 
is  the  total  load  on  one  span: 

Loads  on  Span  Xo. 
123456 

4th  fl.x>r  O         W         W        O 

3rd  floor  W         O        O         W 

2nd  floor  BOWWOW 

1st  floor  W         O         O         W 

With  span  I,  total  story  height  h',  and  moments  of  inertia  I'  of  the  girders 
and  J  of  the  columns  as  before,  the  negative  moment  in  the  second-floor 
girder  between  the  third  and  fourth  span  becomes,  as  may  be  verified  by 
the  slope-deflection  method, 

_  _  / " .  __T  .  . 

(43) 


where  K  =  IJL   as  before    (formula   (29)).     This  moment  is  found  to  be 
approximately  equal  to 

TI7   II  f\   A        \ 

(44) 


When  J  =  oo  the  moment  becomes  Wl/24.  Or,  the  ratio,  Q,  of  increase  of  the 
negative  moment  by  a  change  to  columns  of  finite  stiffness  is  approximately 

Q  =  l  +  T4^  =  l+0.4fc',  (45) 

1  +  K 

where  k'  is  the  freedom  of  the  column  capitals  to  rotate  under  the  loading 
arrangement  in  Fig.  21.  The  ratio  Q  may  be  assumed  to  apply  approxi- 
mately to  the  negative  moments  in  flat  slabs.  For  example,  k'  =  0.72,  as 
given  in  the  first  column  in  Table  5(b),  gives  Q  =  1.29,  or  29  per  cent 
increase,  while  k'  =  0.10  gives  an  increase  of  4  per  cent. 

The  case  in  which  only  a  single  panel  is  loaded  is  of  no  particular 
importance  as  a  condition  for  design;  greater  bending  moments  are  pro- 
duced by  a  uniform  load  on  all  panels  or  by  loading  in  rows  than  by  single- 
panel  loading.  A  number  of  tests  have  been  made,  however,  with  a  single 
panel  loaded,  and  the  case  should  be  investigated  for  the  purpose  of  the 
study  of  these  tests. 


•  See  Wilson,  Richart,  and  Weiss,  Analysis  of  Statically  Indeterminate  Structures, 
by  the  Slope  Deflection  Method,  Univ.  of  111.  Eng.  Exp.  Sta.  Bull.  108,  1918;  Hool 
and  Johnson,  Concrete  Engineers'  Handbook,  1918,  p.  411,  p.  629. 


MOMENTS  AND  STRESSES  IN  SLABS.  51 

When  the  columns  are  stiff,  the  single-panel  load  produces  approxi- 
mately the  same  effects  in  the  panel  as  a  "checker-board  load,"  by  which 
the  panels  corresponding  to  the  black  and  white  fields  of  a  checker-board 
are  loaded  and  unloaded  respectively.*  The  checker-board  load  leads  to  a 
simpler  analysis  than  the  single-panel  load.  The  checker-board  load  W,  0 

may  be  resolved  into  two  components,  one,    .  ..   uniformly  distributed 

9         9 

~\A7  \\f 

over  all  panels,  the  other,  -f -— ,  —  -~,  positive  and  negative  in  alternate 

£  £ 

panels.  The  first  component  produces  one-half  the  moments  derived  in 
Art.  8.  The  second  component,  on  account  of  the  anti-symmetry  leaves  the 
panel  edges  straight;  that  is,  if  the  column  capitals  are  small,  each  panel 
deflects  as  a  single  square  panel  which  is  simply  supported  on  four  sides 
and  loaded  by  +  W/2  or  —  W/2. 

A  study  is  made  of  the  equilibrium  of  one-half  of  a  loaded  panel.  This 
half-panel  is  a  rectangle  with  two  sides  of  length  I,  containing  the  column- 
head  sections,  mid-section,  outer  sections,  and  inner  section,  and  two  sides 
of  length  1/2.  The  moments  are  taken  about  the  edge  containing  the  mid- 
section  and  column-head  sections.  The  following  results  were  found  for  the 
point-supported  slab  with  checker-board  loading  W,  O,  the  moments  being 

expressed  in  terms  of  the  total  moment  M    =  —  Wl: 

Moment  in  inner  section:  0.13A/  ; 

outer  sections:  0.13A/o; 

mid-section:  0.09A/0; 

column-head  section:  0.24A/0. 

Total  for  the  moment  sections:  0.59A/0 

Moments  of  vertical   shears  plus   torsional  moments 

in  the  short  sides  of  the  rectangle:  0.41M 

Total:  \~WM~ 

Moment  of  applied  load  W/2:  _  1.00M0 

Total  moment  about  edge  containing  mid-section :  0. 

According  to  these  results  41  per  cent  of  the  total  moment  leaks  out 
by  shear  and  torsion  in  the  short  edges  of  the  rectangle.  In  deriving  the 
moments  in  the  various  sections  use  was  made  of  the  results,  stated  in 
Art.  8,  Table  3,  obtaiued  for  a  flat  slab  with  all  panels  loaded;  of  the  value 
stated  in  Fig.  3  (a),  of  the  moment  at  the  center  of  a  square  slab  simply 
supported  on  four  sides;  and  of  the  value  of  the  moments  and  shears  at 
other  points  of  a  square  slab  simply  supported  on  four  sides,  stated  by 
Leitz.f  In  slabs  with  column  capitals  the  proportions  of  moments  may  be 
assumed  to  be  approximately  the  same  as  in  the  slabs  with  point  supports, 
provided  the  column  capitals  are  not  too  large.  Or,  by  substituting 

1  *?  r 

M    _  --Wl(l  —  -—)-     one  may  use  the  expressions  just  stated  for  the 
8  of 


*  See  Nielsen,  p.   196,  where  this  case  is  invaitigated. 
t  See  Nielsen,  p.   133. 


52  MOMENTS  AND  STRESSES  IN  SLABS. 

point-supported  slab,  as  approximate  expressions  for  moments  in  the  slab 
with  column  capitals. 

10.  MOMENTS  IN  WALL  PANELS,  CORNER  PANELS,  OBLONG  INTERIOR 
PANELS,  AND  PANELS  WITHOUT  BENDING  KESISTANCE  IN  THE  MID-SECTION. 
Table  VI  contains  approximate  values  of  moments  in  panels  of  several 
different  types.  The  results  of  the  computations  are  found  in  the  column 
next  to  the  last  in  the  table.  The  calculations  are  based  on  Nielsen's  * 
work,  and  they  are  of  an  approximate  character.  In  the  analysis  of  these 
cases  by  difference  equations  Nielsen  used  a  rather  large  value  of  the 
side  X  of  the  elementary  square,  and  he  assumed  point  supports  instead 
of  supports  on  column  capitals.  It  was  his  scheme  that  the  values  deter- 
mined in  this  manner  should  be  used  as  a  basis  of  comparison  between  the 
different  cases;  this  use  has  been  made  of  his  results  in  Table  VI.  The 
exterior  panels  dealt  with  in  Table  VI  are  assumed  to  be  simply  supported 
along  the  walls  on  rigid  lintel  beams. 

An  example  will  illustrate  the  method  followed  in  computing  Table  VI. 
Take  sections  D  and  E  in  the  third  case  in  Table  VI,  that  is,  the  case  of 
two  adjacent  rows  of  wall  panels  with  lintel  beams.  Nielsenf  gives  the 
following  coefficients  M/wl-  of  moment  per  unit  width  when  Poisson's  ratio 
is  zero:  at  the  center  of  D,  at  =  0.0861;  at  the  center  of  E,  Cj  =  0.0661; 
midway  between,  6j  =  0.0725.  The  spaces  between  these  points  are  equal  to 
the  side  \  =  i/4  of  the  elementary  square  which  was  used  in  the  difference 
equations.  The  three  coefficients  are  to  be  interpreted  as  the  altitude  of 
three  rectangles,  placed  together  as  in  Fig.  25  at  the  left.  The  three  rec- 
tangles constitute  the  moment  diagram  in  approximate  form.  A  more 
nearly  correct  diagram  is  obtained  by  drawing  a  smooth  curve  which  has 
the  altitudes  of  the  rectangles  as  average  ordinates  in  the  three  intervals 
covered  by  the  rectangles.  The  curved  diagram  is  now  replaced,  as  shown 
at  the  right  in  Fig.  25,  by  two  rectangles  whose  altitudes  are  average 
ordinates  within  the  two  intervals  covered  by  the  bases.  The  formulas 
stated  in  Fig.  25  for  these  altitudes  apply  approximately,  and  they  were 
used  in  the  calculation  of  the  table.  In  this  manner  the  following  coeffi- 
cients of  moment  per  unit-width  are  found  in  sections  D  and  E : 
in  the  outer  section  D, 

ai  +   bi       Q!   -   ci      0.0861  +  0.0725       0.0861  -  0.0661 

~2~~          ~~15~  ~~2~~  ~T5~~  °6' 

and  in  the  inner  section  E, 

61    +   c,        a.   -   c,       0.0725  +  0.0661        0.0861  -  0.0661 

~T          ~T5~  ~T~  ~l5~ 

The  corresponding  moments  per  unit  width,  O.OS06W  and  0.0680W,  where 
W  =  icV,  and  the  average  value  0.0743W  applying  to  section  F,  are  stated 
in  Table  VI.  By  applying  the  same  method,  with  the  side  of  the  elementary 


*  N.  J.  Nielsen,  Spaendinger  i  Plader,  1920;    the  five  cases  represented  in  Table  6 
are  dealt  with  in  his  work  on  pages  210,  189,  212,  217,  and  205,  respectively, 
t  Nielsen,  p.  214. 


MOMENTS  AND  STRESSES  IN  SLABS. 


53 


TABLE  VI. — MOMENTS  IN  OBLONG  PANELS,  WALL  PANELS,  CORNER  PANELS, 
AND  PANELS  WITHOUT  BENDING  RESISTANCE  IN  THE  MID-SECTION. 

-*  load  per  panel  (uniform  over  a//  panels.).   M0  -B  Wl(i-$  £)* 
'  A  Wld-3  V*.  Ma- 1  Wa(/~§  §)t  Mt-AWtif-J 


Types 
of 
panels 

Sect/on 

Length  of 
secf/on 

Momenf.rln, 
perunifwidfh 
in  Me  Is  en's 
ana/ys/s 
(point  supports, 
A=/^  or  /£) 

Rat  10.  a, 
of/% 
to  the 
Mnfor 
normal 
panel 

Deduced 
MomentM 

f"  Ma  '(J> 

Mf  'moment 
innonTMl 
panel] 

L  oca  t  ion 
of 
sections 

Square  interior 

Normal 

Col-head 
Mid 
Total  ne^ 
Outer 
Inner 
Totalpos 

A 
B 
C 

D 
E 

r 

kl 
zl 
I 
kl 
il 
I 

-O.I067W 
-0.0495W. 
-0.0781  W 
0.057$  W 

6.0469W 

1.00 
I.OO 
I.OO 
1.00 
I.OO 
I.OO 

-0.48M0 
-O.I  7  M0 
-0.65Mo 
O.ZIMo 

JC 

JM1[  III]  1  M  !  1  1   ][  !  *  !  1  11  1  i  '  j  i       JI 

1 

'     1      ^     7_^ 

^-L—  **-  1—  f 
3  'E  |c  F 

Without  bend- 
Ing  resistance 
across  mid 
section 

Cohheact 
Mia 
Outer 
Inner 

A 
B 
D 
E 

\l 
kl 
kl 

1.23 
0.00 
1.10 
129 

-0.59Mo 
0.00 
O.Z3M0 
O./8M0 

Square  exterior 

Two 
adjacent 
rows  of 
wal/ 
pane/s 
with 
lintel 
beams 

Col-head 
Mid 
Jotalneg. 
Outer 
inner 
Tota/pos 
Lintel  ( 
beams\ 
Col-tead 
Mid 

Totalneg. 
Outer 
Inner 

Tofa/pos 

A 

B 
C 
D 

r 

L 
M 

A' 

B: 

r> 

u 

V 

E' 

E; 

F' 

\ 

il 
kl 
I 

il 

I 
il 
kl 

n 
i 

-O.I  380  W 
-0.0646  W 
-0.101  3  W 
0.0806W- 
0.0680W 
0.0743W 
-O.035  W 
0.01  76  W 
-0.1133  W 

-  0.0033  W 
-0.0456W 
0.060/W 
O.O&Z  W 
0.003ZW 
0.0269W 

I.Z9 
I.3O 
I.3O 
1.39 
1.89 
1.58 

1.06 
0.66 

0.58 
1.04 
0.62 

0.57 

-0.22  M^ 

-OMfH> 

6.<55fil 

-0.035WI 
O.OI76WI 
-0.5lr1o 
-O.llMo 
-O.OlMo 
-O.38M0 
02ZM, 

0.09Me 

0.0  1  Mo 
O.SOM* 

III   !  ',    !  I!  '  '  '  ME,    '     :  '[Ilfl 

\ 

-     (. 

-^c— 

~rfc- 

ki""uj      ft       ^ 

li.. 

•.b           J 

Four 
adjacent 
corner 

panels 
with 
lintel 
beams 

Co/.-head 
Mid 

Outer 
Inner 

A 
B 
B, 
D 
E 
E 
d 
e 
f 

i 

unit 
unit 
unit 
unit 
unit 
unit 

-0.1264  W 
-0.0302  W  • 
-0.0008  W 
0.074OW 
0.0446  W 
0.0121  W 
0.0833  W 
0.044-7  W 

0.066  W 
-O.O66   W 

1.18 
O.6I 

I.Z8 
1.24 

-O.57Mo 
-aLOMo 
-Q002Mo 

o.nrio 

0.04  Mo 

f]df)w(?~  7/^1 

.t^rt/ffUr  *Jc/ 

Ehi-SH-g) 

:^A: 

.4-ei 

J     1      ., 

Ob/ong 
inter/or 
panels 
a'=  I  5 

Col-head 
Mid 
Totalnea 
Outer 
Inner 
Totalpos 
Cathead 

Mid 
Tota/n^ 
Outer 
Inner 
Total  pas 

A 
B 
C 
D 
E 
F 
A' 
B' 
C' 
D' 
E 
F 

ka 
ka 
a 
za 

a 

ib 

Zb 
tb 
zb 
b 

-0.1248  wbz 
-0.0314  Wb* 
-0.0781  Wb* 
00696  wbz 
0.0242  wb* 
0.0469  wb* 
-0.10/2  wa* 
-0.0608  wa* 
-0.0810  wa* 
0.0473  WO* 
00407waz 
00440wa* 

W8' 
0.77' 
1.00" 
1.24" 
0.65' 
/.OO" 
0.94' 
1.20* 
I.OO* 
0.90* 
1,16' 
WO* 

-0.52Mb 
-0.13Mb 
-0.65Mb 

o.zeMi, 

0.09Mb 
0.35Mb 
-0.43M, 

-6.66  Ml 

6.  1  6  Ma 

i^  ^P  -A® 

T  r? 
]Wjj 

*  Ratio  of  M  to  tneM  for  norma/ pane/  w/fti  2*0  or  I 


54 


MOMENTS  AND  STRESSES  IN  SLABS. 


square  again  equal  to  Z/4,  to  the  normal  interior  panel,  one  finds  the 
moment  in  the  outer  section  D  equal  to  0.0579  W.  The  ratio  of  increase  for 
the  wall  panel  is  calculated,  then,  as  q  =  0.0806/0.0579  =  1.39  (see  fifth 
column  for  q).  If  this  ratio  is  correct,  the  moment  in  section  D  in  the 
wall  panel,  when  the  columns  are  replaced  by  point  supports,  may  be  com- 
puted as  M  "  qMq  where  M  is  the  moment  in  section  D  in  the  normal 

panel,  that  is,  M  =  1.39  '0.21  •  ¥1  «  0.29  •  — Z-  When  the  point  supports 

w  i 
are  replaced  by  columns  with  capitals,  one  may  expect  that  the  factor 

1  1    r  1 

is  to  be  replaced  by    M'0  =  -  Wl  (1  —  _  1)2.    the  coefficient    i  occurs  in 

8  3  I  3 

o 
this  expression  instead  of  the  usual  because  the  clear  span  in  this 

o 
case  is  I  —  %c  instead  of  the  value  I  —  c  found  in  the  interior  panels. 

The  moments  in  the  lintel  beam  in  sections  L  and  M  are  computed 
under  the  assumption  that  the  beam  is  continuous,  with  supports  opposite 
the  columns. 


^v 

< 

1  —  = 

T 

\ 

X, 

\ 

r 

-Q 

'    '•  —  

~—  - 

g+b+d^c          -T-—     

'     2.       is           \b+c     a-c 

I 

<j 

z        is 

i 

i 

i  i 

FIG.  25. — DIAGEAM  INDICATING  RELATIONS  BETWEEN  AVEBAGE  OKDINATES. 

The  values  stated  for  panels  without  bending  resistance  across  the  mid- 
section  were  estimated  on  the  basis  of  a  similar  case  treated  by  Nielsen,  in 
which  there  is  no  bending  resistance  in  four-fifths  of  the  mid-section.  The 
proportions  are  nearly  equal  to  those  given  by  Nielsen.  The  side  of  the 
elementary  square  used  in  this  case  was  \  =  1/5. 

The  oblong  interior  panels  have  sides  a  and  &,  with  a  =  1.55.  The 
elementary  squares  used  by  Nielsen  had  the  side  \  =  6/4  =  a/6-  The 
moments  M  in  the  short  sections  A',  B',  D',  and  E'  were  determined  in  the 
manner  described  in  sections  D  and  E  in  the  wall  panels.  The  average 
moment  per  unit-width  in  each  of  the  sections  A,  B,  D,  and  E  was  deter- 
mined as  the  average  of  three  values  given  by  Nielsen,  one  for  each  ele- 
mentary square  adjacent  to  the  section.  The  proportions  of  negative  and 
positive  moments  M  found  in  this  way  are  approximately  the  same  as  in 
the  normal  panel.  If  they  can  be  assumed  to  be  exactly  the  same,  the 
total  moments  in  the  sections  C,  C',  F,  and  F'  become  _065A/6,  —  0.65A/a, 
0.35Af  ,  and  0.353/  ,  respectively,  as  stated  in  the  column  for  M.  These 
total  moments  are  assumed  to  be  distributed  between  the  column-head  sec- 
tions and  mid-sections,  outer  sections  and  inner  sections  according  to  the 


MOMENTS  AND  STRESSES  IN  SLABS. 


55 


proportions  between  the  values  of  Mn,     For  example,  one  finds  under  this 
assumption  in  the  column-head  section: 

-  0.1248**.. L      (_ 


M  - 


0.52M, 


-  0.0781u>&2  •  a  v  -.~wb)  -  v,.u^«ft. 
The  remaining  values  of  M  were  computed  in  the  same  manner.  The 
ratios  of  M  to  the  corresponding  M  in  the  normal  panel  were  computed 
afterward. 

The  moments  in  the  oblong  panels  are  represented  graphically  in  Fig. 
26,  where  the  abscissas  are  values  of  the  ratio  V*i  of  the  spans,;  Z,  is  the 
span  in  the  direction  in  which  the  moment  is  taken.  The  points  represent- 
ing the  moments  in  Table  VI  lie  on  or  near  the  two  straight  lines  and  the 


-45 


I* 

\30 
$ 

V* 

*1 

!/5 
£/* 


—  ^lu&g&L 


ms.tt%£ 


VTufcl 


_f^f^^ 

r&&M^~ 


oiz 


0.4 


i.o 


O.6  O.8 

Ratio  of  Spans, 

FIG.  26. — MOMENTS  IN  OBLONG  PANELS. 


iz 


1.4 


two  parabolas  which  are  shown,  with  their  equations,  in  Fig.  26.  These 
equations  may  be  used  as  formulas  for  the  moments  in  the  various  moment 
sections  in  oblong  panels.  They  show,  as  might  be  expected,  that  the 
moments  are  more  nearly  uniformly  distributed  over  the  short  sections 
than  over  the  long  sections. 

Nielsen  derived  the  following  values  of  the  reactions  of  the  columns: 
in  the  case  of  two  adjacent  rows  of  wall  panels,  R  =  1.203W;  in  the  case 
of  four  adjacent  corner  panels,  R  =  1.313W. 

Nadai  *  derived  the  following  values  of  the  moments  per  unit  width 
for  a  single  square  panel,  which  is  simply  supported  on  point  supports  at 
the  four  corners;  he  assumed  Poisson's  ratio  equal  to  0.3;  at  the  center  of 
the  panel  3/ =  0.1115W,  at  the  center  of  the  edge  in  the  direction  of  the 
edge,  M  =  0.151TF,  the  latter  being  the  maximum  moment  in  the  panej. 


•  NAdai,  p.  70. 


56  MOMENTS  AND  STRESSES  IN  SLABS. 

III.— RELATION  BETWEEN  OBSERVED  AND  COMPUTED  TENSILE 
STRESSES  IN  REINFORCED  CONCRETE  BEAMS. 

BY  W.  A.  SLATER. 

11.  GENERAL  CONSIDERATIONS.    It  has  been  generally  recognized  that  in 
the  tests  which  have  been  made  on  reinforced-concrete  floors  the  measured 
tensile  stresses  in  the  reinforcement  do  not  account  for  all  of  the  moments 
which  are  applied  to  the  slab.     This  has  been  especially  apparent  in  the 
cases  in  which  the  measured  stresses  were  low.    In  the  tests  of  flat  slabs  the 
coefficient  of  the  resisting  moment  of  the  measured  steel  stresses  has  been 
found  to  increase  as  the  measured  stresses  increased.    This  increase  in  the 
coefficient   indicates   that,   for   the   low   loads   at   least,   the   tensile   stress 
accounts  for  only  a  portion  of  the  applied  bending  moment.     Table  VII 
quotes  published  results  which  show  that  for  different  loads,  the  difference 
in   the  proportion   of  the   total  moment  which   is   accounted  for   by  the 
measured  tensile  stresses  is  likely  to  be  considerable.     It  is  very  desirable 
that  a  means  be  found  for  determining  how  great  this  difference  is,  but  the 
slab  tests  cannot  be  used  for  this  purpose  since  the  coefficient  of  the  bending 
moment  is  the  main  thing  that  is  in  question  in  the  slab  tests. 

To  show  the  relation  between  the  bending  moment  and  the  resisting 
moment  in  beams  a  study  of  observed  tensile  stresses  in  84  reinforced- 
concrete  beams  tested  at  an  age  of  13  weeks  has  been  made.  These  beams 
were  tested  by  the  Technologic  Branch  of  the  United  States  Geological 
Survey  at  St.  Louis  about  1905  to  1908,  and  have  been  reported*  in  Techno- 
logic Paper  No.  2  of  the  Bureau  of  Standards,  by  Humphrey  and  Losse. 
All  the  test  data  of  those  beams  here  presented  may  be  found  in  that  paper. 
The  results  of  this  study  are  shown  in  Fig.  31  and  32  as  a  relation  between 
the  observed  and  the  computed  stresses,  but  it  is  evident  that  the  relation 
of  the  observed  stress  to  the  computed  stress  is  the  same  as  the  relation  of 
the  computed  moment  of  the  observed  tensile  stresses  to  the  applied  moment. 
Test  results  from  the  same  source  and  from  other  sources  were  used  by 
Professor  Hatt  in  a  study  of  the  relation  of  the  computed  moment  of  the 
measured  tensile  stresses  in  the  reinforcement  to  the  applied  moment  for 
beams  and  slabs.  Professor  Hatt  found  that  beam  tests  from  different 
sources  gave  different  results,  and  the  writer  also  has  found  this  to  be  true. 
It  is  shown,  however,  in  Professor  Hatt's  paperf  that  there  is  a  certain 
degree  of  conformity  between  test  results  from  a  considerable  variety  of 
sources,  and  the  writer  feels  justified  in  using  the  beams  of  Technologic 
Paper  No.  2  for  the  purpose  of  determining  a  law  which  will  serve  as  a 
basis  for  the  comparative  study  of  the  moment  in  slabs. 

12.  LIMITATIONS  TO  GENERAL  APPLICABILITY  OF  THE  TEST  RESULTS 
While  it  is  recognized  that  there  are  limitations  which  must  be  observed  in 
the  application  of  these  results  to  other  conditions  than  those  under  which 
they  were  obtained,  it  is  believed  that,  because  of  the  wide  range  which  the 


•  See  alsn  Bulletins  329  and  344,  U.  S.  Geological  Survey. 

t  W.   K.   Hatt.   Moment  Coefficients  for  Flat   Slab  Design  with  Results  of  a  Test 
Proceedings  American  Concrete  Institute,  Vol.   14,  p.   165   (1918). 


MOMENTS  AND  STRESSES  IN  SLABS. 


57 


tests  cover,  the  limitations  are  less  serious  than  would  be  those  of  any  other 
tests  which  might  have  been  used  for  this  purpose. 

For  the  beam  tests  there  are  reasons  for  expecting  a  difference  between 
the  observed  stresses  and  the  computed  stresses.  In  a  cracked  beam  the 
stress  at  the  cracks  may  approach  the  computed  stress,  but  between  the 
cracks  the  concrete  assists  so  greatly  in  carrying  the  stresses  that  the 
average  measured  unit-deformation  over  the  gage  length  is  likely  to  be 
considerably  less  than  the  maximum  unit-deformation,  especially  at  the 
lower  loads.  It  is  possible  also  that  even  at  the  section  where  a  crack 
occurs  a  portion  of  the  moment  may  be  resisted  by  the  tensile  stresses  in  the 
concrete.  There  probably  are  other  reasons  for  the  differences  though  it  is 
believed  that  these  are  the  most  important. 

By  taking  measurements  of  deformation  over  a  short  gage  length 
within  a  longer  gage  length  on  the  same  reinforcing  bar  it  has  been  shown* 
that  higher  stresses  exist  at  certain  places  than  are  indicated  by  the  average 
measured  deformation.  The  fact  that  in  the  beams  used  as  the  basis  of  the 
present  study  failure  occurred  at  observed  stresses  which  were  somewhat 

TABLE  VII. — UNCORRECTED  MOMENT  COEFFICIENTS  FROM 
PUBLISHED  REPORTS. 

(Sum  of  coefficients  for  positive  moment  and  negative  moment.) 


Test. 

Load, 
Ib.  per 

Coef- 
ficient. 

Load, 
Ib.  per 

Coef- 
ficient. 

Load, 
Ib.  per 

Coef- 
ficient. 

Load, 
Ib.  per 

Coef- 
ficient. 

Reference. 

sq.  ft. 

sq.  ft. 

sq.  ft. 

sq.  ft. 

Shredded  Wheat  Factory 

56 

0  0363 

120 

0  0356 

191 

0  0439 

Proc.  A.  C.  I.,  1914, 

p.  404. 

Worcester  Test  Slab 

102 

0  026 

215 

0  049 

Univ.  of  Illinois  Eng. 

Exp.  Sta.  Bull.  84, 

p.  103. 

Purdue  Test,  "J"  Slab.. 

150 

0.0148 

300 

0.0246 

450 

0.0400 

600 

0.0539 

Proc.  A.  C.  I.,  1918, 

p.  181. 

lower  than  the  known  yield  point  of  the  steel  is  another  indication  that  even 
at  the  high  loads  the  real  stresses  were  greater  than  the  observed  stresses. 

There  is  another  limitation  to  the  interpretation  of  test  results  ot 
slabs,  which  the  results  of  the  study  of  the  beam  tests  does  not  help  to 
remove.  This  limitation  lies  in  the  fact  that  in  the  measurement  of  tht 
deformation  in  a  slab  at  a  position  of  rapidly  changing  stress  the  result 
represents  the  average  unit  deformation  over  the  entire  gage  length  and 
not  at  the  end  where  the  stress  in  the  gage  length  is  a  maximum.  In  the 
beams  tested  the  moment  was  constant  over  the  entire  gage  length. 

13.  TEST  SPECIMENS  AND  METHODS  OF  TESTING.  Technologic  Paper 
No.  2f  of  the  Bureau  of  Standards  describes  the  specimens  and  tho 
methods  of  testing.  Only  the  features  which  arc  of  tJio  most  importance  in 
connection  with  the  present  study  arc  repeated  hero. 


*  F.   R.   McMillan,   in  unpubli.-lied  data   furnished  to  the  writer, 
t  See  also  Bulletin  329,  U.  S.  Geological  Survey. 


58 


MOMENTS  AND  STRESSES  IN  SLABS. 


All  the  beams  were  13  ft.  long,  8  in.  wide,  and  11  in.  deep.  The 
amount  of  reinforcement  varied  in  the  different  beams  from  two  one-half- 
inch  round  to  eight  one-half-inch  round  bars.  For  the  beams  having  four 
bars  or  less  all  were  placed  in  one  layer.  For  the  beams  having  five  bars 
or  more  the  bars  were  placed  in  two  layers.  The  vertical  distance  between 
the  centers  of  the  bars  of  the  two  layers  was  l1/^  in.  The  distance  from  the 
top  of  the  beam  to  the  center  of  the  lover  layer  of  bars  was  10  in.  The 
ratio  of  reinforcement,  based  upon  the  depth  to  the  center  of  gravity  of 
the  reinforcement,  varied  in  the  different  beams  from  0.0049  for  the  beams 
with  two  bars,  to  0.0212  for  the  beams  with  eight  bars.  In  Technologic 
Paper  No.  2  the  percentage  of  reinforcement  was  based  upon  the  depth  to 
the  center  of  the  lower  layer  of  bars.  On  account  of  the  difference  in  the 
method  of  computing  the  ratios  of  reinforcement  the  ratios  given  here  are 
slightly  greater  for  the  beams  having  two  layers  of  bars  than  the  ratios 
given  in  Technologic  Paper  No.  2. 

All  the  concrete  used  in  the  beams  was  of  a  1:2:4  mixture.  Four 
different  aggregates  were  used.  These  were  granite,  gravel,  limestone,  and 

TABLE  VIII. — STRENGTH  MODULUS  OF  ELASTICITY  AND  VALUES  OF  n. 


Compressive 
Strength 
!b.  per  sq.  in. 

Modulus  of 
Elasticity, 
Ib.  per  sq.  in. 

Values  of  n. 

4,200 

4,430,000 

6.8 

4,000 

4,620,000 

6.5 

Limestone  Concrete  ..            ...          .... 

3,600 

3,810,000 

7.9 

Cinder  Concrete  

2,200 

1,820,000 

16.6 

NOTB. — The  term  n  represents  the  ratio  of  the  modulus  of  elasticity  of  the  steel  to  that  of  the  concrete. 
For  this  computation  the  modulus  of  elasticity  of  the  steel  is  taken  at  30,000,000  Ib.  per  sq.  in. 

cinders.  The  strengths  and  moduli  of  elasticity  of  the  concretes  from  these 
aggregates  are  given  in  Table  VIII.  It  will  be  seen  that  while  all  of  the 
concrete  had  a  good  strength  there  was  considerable  range  in  the  strengths. 

All  the  bars  used  as  reinforcement  in  these  beams  were  tested  and 
they  were  classified  according  to  the  yield  point.  (In  the  earlier  beams 
the  elastic  limit  was  used  as  the  basis  of  classification.)  This  insured 
practically  the  same  yield  point  for  all  the  bars  of  any  beam.  The  highest 
and  the  lowest  yield  points  for  the  beams  quoted  in  this  paper  were  43470 
and  36110  Ib.  per  sq.  in.  respectively.  The  unit  used  in  the  discussion  in 
this  paper  is  the  average  for  three  beams  of  a  kind.  On  this  basis  the 
highest  and  lowest  average  yield  points  were  42900  and  36400  Ib.  per  sq.  in. 
respectively.  The  average  yield  point  was  40200  Ib.  per  sq.  in.  The  yield 
points  for  all  but  two  of  the  twenty-eight  groups  were  within  o1/^  per  cent 
of  the  average. 

A  span  of  12  ft.  was  used  in  all  the  tests,  and  the  loads  were  applied  at 
the  one-third  points  of  the  span.  The  deformations  were  measured  over  a 
gage  length  of  29.25  in.  by  means  of  extensometers  which  were  attached  to, 


MOMENTS  AND  STRESSES  IN  SLABS.  59 

the  concrete  and  which  did  not  measure  directly  the  deformations  in  the 
steel.  The  arrangement  of  the  extensometers  gave  the  deformations  in  the 
steel  at  the  level  of  the  bottom  layer  of  reinforcement.  These  are  the 
deformations  on  which  the  studies  in  the  Technologic  Paper  are  based. 
Since  in  the  present  study  the  depth  of  the  beam  was  taken  as  the  depth 
to  the  center  of  gravity  of  the  cross  section  of  the  reinforcement  it  became 
necessary  to  reduce  the  deformations  reported  in  the  Technologic  Paper  to 
the  corresponding  deformations  at  that  depth.  This  modification  affected 
only  the  beams  having  more  than  one  layer  of  bars. 

14.  METHOD  OF  ANALYZING  RESULTS  OF  BEAM  TESTS.  The  analysis  of 
the  results  of  the  beam  tests  consists  in  deriving  an  empirical  equation  for 
the  observed  stress  in  terms  of  the  computed  stress  and  the  percentage  of 
reinforcement.  For  the  beams  used  in  this  study  the  load-strain  diagrams 
(in  which  values  of  H/bd?  were  plotted  as  ordinates  and  the  unit- 
deformations  in  the  steel  were  plotted  as  abscissas)  are  made  up  of  three 
parts,  ( 1 )  the  part  in  which  little  or  no  cracking  of  the  concrete  had  taken 
place;  (2)  the  part  in  which  the  concrete  had  cracked  and  the  stress  in  the 
reinforcement  was  below  the  yield  point;  and  (3)  the  part  in  which  the 
stress  in  the  steel  was  at  or  beyond  the  yield  point.  It  was  found  that  the 
first  two  parts  were  nearly  straight  lines  which,  if  projected,  intersected 
at  a  point  which  corresponds  quite  closely  to  the  unit-deformation  at  which 
a  breaking  down  of  the  concrete  in  tension  may  be  expected  to  occur.  In 
the  study  of  the  results  of  these  beam  tests  empirical  equations  for  these 
two  straight  lines  were  determined.  In  the  diagrams  representing  these 
empirical  equations  the  straight  lines  are  connected  by  smooth  curves,  but 
no  attempt  has  been  made  to  state  an  equation  for  the  curved  portion.  The 
part  of  the  load-strain  diagram  for  which  the  yield  point  of  the  steel  has 
been  reached  or  passed  has  not  been  included  in  the  study. 

Fig.  27  gives  typical  load-strain  diagrams*  for  certain  of  the  gravel 
concrete  beams  used  in  this  study.  In  Fig.  29  there  is  a  sketch  of  a  load- 
strain  diagram  with  notation  which  will  assist  in  making  clear  the  manner 
in  which  the  analysis  was  carried  out.  The  lines  OA  and  BC  represent  the 
straight  lines  which  may  be  fitted  to  the  two  portions  of  the  diagrams 
below  the  yield  point  of  the  steel.  The  slopes  of  the  line  OA  for  all  the 
beams  used  were  plotted,  and  from  these  points  an  equation  for  the  slope 
was  derived.  The  plotted  points  and  the  equations  of  the  lines  which  were 
fitted  to  them  are  given  in  Fig.  28.  Likewise  the  slopes  of  the  lines  BC  for 
all  the  beams  used  were  plotted  in  Fig.  29  and  the  equation  of  the  slope  was 
derived.  The  intercept  OB  of  the  lines  BC  for  all  the  beams  were  plotted 
in  Fig.  30  and  equations  of  the  intercept  were  derived  in  like  manner.  The 
height  of  the  intercept  OB  might  be  used  as  a  measure  of  the  load  at  which 
the  breaking  down  of  the  concrete  in  tension  occurred,  but  it  is  not  entirely 
satisfactory  as  a  measure  of  this  action  of  the  beams,  and  Fig.  30,  showing 
these  intercepts  plotted  as  ordinates,  is  introduced  only  to  indicate  this 


•  The  tabulated  data  used  in   this  figure   are  given  in   Tech.   Paper  No.   2,  U.   S. 
Bureau  of  Standards,  1911. 


60 


MOMENTS  AND  STRESSES  IN  SLABS. 


Strain 


Deflection 


Neutral  Axis 


~4 


'§100 

<^ 

^ 

^    0 

M 
ffl 

100 
0 

300 
200 
100 


Beam  Na34d 


^7/77 


3-k"rd  rods-074% 


i 


» 


Deflect/on 


I 


Q\5 


' 


:• 


06 


07 


_L 


01 


oz 


FIG.  27.- --TYPICAL  LOAD-STRAIN   DIAKKASUS  FOR  BEAM  TESTS. 


MOMENTS  AND  STRESSES  IN  SLABS. 


61 


step  in  the  derivation  of  the  equations  of  the  relation  between  the  observed 
and  the  computed  stresses.  The  equation  of  the  slopes  of  the  lines  OA,  and 
the  fact  that  the  lines  pass  through  the  origin  give  sufficient  information 
with  which  to  determine  the  equation  of  the  lines  OA.  The  equation  of  the 
slope  of  the  lines  BC,  and  the  equations  of  the  intercept  OB  of  these  lines 
on  the  vertical  axis,  were  stated  in  terms  of  observed  and  computed  stresses, 
and  were  solved  simultaneously  for  the  equations  of  the  lines  BC.  The 
graphical  representation  of  the  equations  determined  in  this  way  for  the 
granite,  gravel,  and  limestone  concretes  is  given  in  Fig.  31.  The  equations 
which  apply  to  the  beams  of  cinder  concrete  are  somewhat  different  and 
are  shown  in  Fig.  32.  In  all  the  computations  the  value  of  /  (the  ratio  of 
the  moment  arm  to  the  depth  d)  was  taken  as  0.86G. 


kj  2500000 

=§  zoooooo 

fc  1500000 
•*^ 
^ 
^  1000000 

^n 

v-QmrirteCOt 
o  Gravel 
•Limestone 
^Cinder 

x/gvfe 

O   i  —  •" 

0 

^-* 

o 

'0000- 

o 

fJ^W; 
& 

V°°l\ 

^~  ' 

'         0 

"~~~  « 

.^  

I 

• 

~~~~~ 

—  •  — 

X 

""  • 

—  —  ' 

V 

• 

.^  

oo+z 

-i».  — 

soooa 

—  +~ 

7P 

,     — 

..        -* 

—4— 

—  —  r" 

~* 
+ 

-T 

^—  -t— 
M 
~e5d* 

» 
-^252 

1  i»j  jUUOOO 

L 
FIG..  28.—  Ri 

LOAD-S' 

?    .<aa?  .##  .#»  .-006  .010   .o/z  .014  .0/6  .o/d   020  .022 
Faf/o  of/?einforcemenf,p 

:LATION  BETWEEN  RATIO  OF  REINFORCEMENT  AND  SLOPE  OF 
CHAIN  CURVE  BELOW  LOAD  AT  WHICH  CONCRETE  CRACKED. 

15.  EFFECT  OF  QUALITY  OF  CONCRETE  AND  AMOUNT  OF  REINFORCEMENT 
ON  RATE  OF  INCREASE  OF  TENSILE  DEFORMATION.  Fig.  28  and  29  were 
prepared  in  order  to  study  the  action  of  the  beam  during  the  first  two 
stages  of  the  test  which  have  been  referred  to  in  Art.  14.  In  each  figure  the 
ordinates  of  the  plotted  point  represent  slopes  of  a  portion  of  the  load-strain 
diagrams  for  all  the  beams  used  in  this  study.  These  slopes  may  be  inter- 
preted as  the  rate  of  increase  of  deformation  with  load.  Fig.  28  represents 
the  stage  of  the  test  below  the  cracking  of  the  concrete.  Fig.  29  represents 
the  stage  of  the  test  between  the  cracking  of  the  concrete  and  the  reaching 
of  the  yield  point  of  the  reinforcement. 

For  stages  of  the  test  below  the  cracking  of  the  concrete  the  rate  of 
increase  of  tensile  deformation  was  affected  in  an  important  degree  by  the 
quality  of  the  concrete,  while  the  effect  of  the  amount  of  reinforcement  on 
the  rate  of  increase  of  deformation  was  almost  negligible.  For  stages  of 
the  test  above  the  cracking  of  the  concrete  the  conditions  were  reversed; 


62 


MOMENTS  AND  STRESSES  IN  SLABS. 


the  rate  of  increase  of  tensile  deformation  was  affected  in  an  important 
degree  by  the  amount  of  reinforcement,  while  the  effect  of  the  quality  of 
the  concrete  on  the  rate  of  increase  in  deformation  was  entirely  negligible. 
In  Fig.  28  the  magnitudes  of  the  ordinates  to  the  respective  points  are, 
in  general,  in  the  order  of  the  values  of  the  modulus  of  elasticity  of  the 
concretes .  represented.  The  ordinates  representing  the  cinder  concrete, 
which  had  a  modulus  of  elasticity  about  half  as  great  as  that  of  the  other 
concretes  (see  Table  VIII),  are  uniformly  about  half  of  those  for  the  other 


5500OO\ 


OOZ    004     006    .006     OIO    .0/Z     .0/4     0/6 

ffbtio  of  Reinforcement,   p 


0/8     020    022 


FIG.    29. — RELATION   BETWEEN   RATIO   OF   REINFORCEMENT   AND   SLOPE   OF 
LOAD-STRAIN  DIAGRAM  ABOVE  LOAD  AT  WHICH  CONCRETE  CRACKED. 


concretes.  Even  for  the  granite,  gravel,  and  limestone  concretes,  whose 
moduli  of  elasticity  showed  only  slight  differences,  the  magnitudes  of  the 
average  ordinates  to  the  points  take  the  same  order  as  the  values  of  the 
modulus  of  elasticity.  Although  the  compressive  strengths  occupy  the  same 
order  of  magnitude  as  the  moduli,  it  seems  logical  to  attribute  the  effect 
on  the  rate  of  deformation  to  the  variation  in  the  modulus  of  elasticity 
rather  than  to  the  variation  in  the  compressive  strength.  Whether  the 
important  factor  was  the  modulus  of  elasticity  or  the  compressive  strength, 
the  facts  here  pointed  out  justify  the  statement  that  the  rate  of  tensile 


MOMENTS  AND  STRESSES  IN  SLABS. 


6.3 


deformation  was  affected  in  an  important  degree  by  the  quality  of  the 
concrete. 

Since  the  abscissas  in  Fig.  28  are  ratios  of  reinforcement  the  slopes  of 
the  curves  fitted  to  the  plotted  points  in  this  figure  will  be  a  measure  of  the 
effect  of  the  amount  of  reinforcement  on  the  rate  of  increase  of  tensile 
deformation.  Both  average  lines  (that  for  stone  and  gravel  concretes  and 
that  for  cinder  concrete)  have  slopes  which  are  very  small  in  proportion 
to  the  slope  of  the  line  in  Fig.  29,  whicli  represents  the  conditions  for  the 
stage  of  the  test  after  the  formation  of  cracks.  This  comparison  justifies 
the  statement  that  the  effect  of  the  amount  of  reinforcement  on  the  rate 
of  deformation  was  almost  negligible  for  the  stage  of  the  test  in  which  the 
concrete  was  not  generally  cracked. 

The  greatly  increased  slope  of  the  average  line  in  Fig.  29  over  the 
slopes  shown  in  Fig.  28  forms  the  basis  of  the  statement  that  for  stages 

•%Z50 


¥i=f?o  +(  '0-002)3000 
bdz         ,       / 

• 

_—* 

_—  ' 

-*  —  • 

-r 

"•' 

-" 

—  ox-  — 

—  «-{—  ~° 

+ 

-t- 

I    + 

/* 

+ 

z* 

xGra 
06  m 
•  Lm 
+Cin(. 

nite  Concrete. 
'el      ' 
tsfone.  • 
1er 

§ 

,8 

-5      0       .002    .004     .006     .003    .O/O     .0/2    .014     .0/6     .0/8    .O2O   .022 

ffistio  of  fcs/nforcemenf,  p=-£j 

FIG.   30. — RELATION   BETWEEN  RATIO  OF  REINFORCEMENT  AND  INTERCEPT 
OB  OF  FIG.  29  ON  LOAD  Axis. 


of  the  test  above  the  cracking  of  the  concrete  the  rate  of  tensile  deformation 
was  affected  in  an  important  degree  by  the  amount  of  reinforcement.  To 
appreciate  fully  the  relative  importance  of  this  factor  account  must  be 
taken  of  the  fact  that  in  Fig.  29  the  scale  of  ordinates  is  only  one-tenth  of 
that  used  in  Fig.  28. 

In  the  arrangement  of  the  points  in  Fig.  29  there  is  no  regularity 
which  is  dependent  upon  the  strength  or  modulus  of  elasticity  of  the  con- 
crete. The  points  representing  the  cinder  concrete  are  intermingled  with 
the  points  representing  the  other  grades  of  concrete  in  such  a  way  that  one 
average  line  represents  all  the  results  with  as  great  accuracy  as  would  be 
possible  with  an  independent  line  for  each  kind  of  concrete.  This  clearly 
warrants  the  previous  statement  that  for  the  stages  of  the  test  above  which 
cracks  occurred  the  effect  of  the  quality  of  the  concrete  on  the  rate  of 
tensile  deformation  was  entirely  negligible. 


64 


MOMENTS  AND  STRESSES  IN  SLABS. 


16.  RELATION  BETWEEN  OBSERVED  AND  COMPUTED  TENSILE  STRESSES. 
The  significance  and  scope  of  the  results  of  the  study  of  the  relation 
between  the  observed  and  the  computed  stresses  in  the  reinforcement  may 
best  be  visualized  by  reference  to  Fig.  31  and  32,  which  show  graphically  the 
derived  equations  of  the  observed  stress  in  terms  of  the  computed  stress 
and  the  percentage  of  reinforcement.  For  the  part  of  the  test  below  which 
the  concrete  is  cracked  these  equations  are, 

0.52/s 


1  + 


.021 
P 


1.04f 


for  the  stone  and  gravel  concretes  and 


for  the  cinder  concrete. 


(1) 


(2) 


For  max.  load  J C-400OO^  40000(02^ 


10000     20000    3OOOO    40000    50000    6OOOO    7QOOO 

Computed  5fresj,  £,  /b  per  54.  in 

FIG.  31. — RELATION  BETWEEN  OBSERVED  AND  COMPUTED  TENSILE  STRESSES 
IN  REINFORCEMENT  OF  BEAMS  TESTED  AT  ST.  Louis. 

Diagrams    from    derived    equations    for    beams    of    stone    concrete    and    of    gravel 
concrete's. 

For  the  part  of  the  test  above  which  the  concrete  is  generally  cracked  the 
equations  are, 

1.04p/_  -  144 


p  -  .002 
1.04p/_  -  144 


—  3600  for  stone  and  gravel  concretes  and, 


for  cinder  concrete. 

p  -  .002 


(3) 


(4) 


In  these  equations,  /  is  the  observed  stress,  /      is  the  computed  stress,  and 
p  is  the  ratio  of  longitudinal  reinforcement  based  upon  the  depth  from  the 


MOMENTS  AND  STRESSES  IN  SLABS. 


65 


compression  surface  of  the  beam  to  the  center  of  gravity  of  the  tension 
reinforcement. 

In  Fig.  31  and  32  the  lines  representing  the  stresses  in  beams  having 
less  than  0.5  per  cent  of  reinforcement  are  dotted  because  these  lines  repre- 
sent extrapolation  below  the  lowest  percentage  of  reinforcement  used  in  any 
of  the  beams  tested.  Since  most  of  the  slabs  to  whose  study  the  results  of 
these  beam  tests  may  be  applied  have  not  more  than  0.5  per  cent  of  rein- 
forcement it  is  important  to  consider  whether  the  extrapolation  is  justifi- 
able. The  average  lines  fitted  to  the  points  in  Fig.  28,  29  and  30  were 
projected  from  0.0049,  the  lowest  ratio  of  reinforcement  for  any  of  the 
beams  tested,  to  the  lower  value  of  0.002.  This  extrapolation  covers  a 
small  portion  of  the  total  range  in  the  percentages  of  reinforcement  repre- 


4000C 


for  max. 


/Mdf-4tXm£-400G0(8Z+7p)-y&/^-r--. 
) I    /£-  X//1/   / 


"ffiZzoodT 


o          loooo       zoom      30000       40000      50000      soooo       TOOOO 
Gompufed  Stress,  g,  Ib.  per  sq/n 

FIG.  32. — RELATION  BETWEEN  OBSERVED  AND  COMPUTED  TENSILE  STRESSES 
IN  REINFORCEMENT  OP  BEAMS  TESTED  AT  ST.  Louis. 

Diagrams  prepared  from  derived  equations  for  beams  of  cinder  concrete. 

sented  by  the  beams  which  were  tested,  and  the  curves  which  were  projected 
are  well  denned  by  the  experimental  points.  It  seems,  therefore,  thstt  there 
is  justification  for  this  extrapolation.  An  inspection  of  Fig.  28,  29  and  30, 
on  which  the  curves  of  Fig.  31  and  32  are  based,  will  assist  in  forming  an 
opinion  as  to  whether  the  extension  of  the  scope  of  the  diagrams  of  Fig.  31 
and  32  below  0.49  per  cent  of  reinforcement  is  warranted  by  the  test  data. 
Fig.  29,  31,  and  32  indicate  that  for  beams  having  only  0.2  per  cent 
of  reinforcement  when  the  concrete  breaks  down  in  tension  the  reinforce- 
ment immediately  would  be  stressed  to  failure.  That  is,  when  the  concrete 
breaks  down  in  tension  the  slope  of  the  load  strain  diagram  becomes  zero 
for  a  beam  with  only  0.2  per  cent  of  reinforcement.  That  this  condition 
ia  approached  as  the  amount  of  reinforcement  becomes  small  is  shown  by 


66 


MOMENTS  AND  STRESSES  IN  SLABS. 


inspection  of  Fig.  29.  The  same  thing  is  shown  directly  in  the  flatness 
of  the  slope  of  the  load  stress  diagrams  for  beams  336,  337  and  338  of 
Fig.  27,  which  have  the  smallest  amount  of  reinforcement  of  any  of  the 
beams  studied.  That  there  should  appear  to  be  no  difference  in  the  amount 
of  reinforcement  required  to  bring  about  this  condition  for  the  beams  of 
cinder  concrete  from  that  which  was  required  for  the  beams  of  stone 
concrete,  may  be  due  to  a  break  in  the  mean  line  of  Fig.  29  between  0.5 
and  0.2  per  cent  of  reinforcement.  Whether  such  a  break  occurs  is  not 


40000 


^30000 


*  Granite  Concrete 

o6ra/ef 

•Limestone 


zoooc 


10000 


10000         20000       3000O       4OOOO      5OOOO 
Computed  Stress  -  Ib.  per  53.  //?. 

FIG.  33. — RELATION  BETWEEN  OBSERVED  AND  COMPUTED  TENSILE  STRESSES 
IN  REINFORCEMENT  OF  BEAMS  TESTED  AT  ST.  Louis. 

Test   results   for   beams   of   stone   concrete    and   of   gravel    concrete   compared   with 
results    from    derived    equations. 

known  because  no  beams  with   less  reinforcement   than  0.49  per  cent  were 
tested. 

In  order  to  make  certain  by  a  direct  comparison,  that  equations  ( 1  )  to 
(4)  represent  the  relation  between  the  observed  and  the  computed  tensile 
stresses,  Fig.  33  and  34  have  been  prepared.  Points  showing  the  observed 
and  the  computed  unit  deformations  throughout  the  tests  of  representative 
beams  have  been  plotted,  and  for  comparison  with  them  the  graphs  of  the 
equations  which  represent  the  relation  between  the  observed  and  the 
computed  stresses  for  the  same  beams  are  shown  in  the  same  figures. 
Each  point  plotted  in  these  figures  represents  the  average  load  and  the 
average  deformation  for  the  three  beams  of  its  kind.  Considering  the 


MOMENTS  AND  STRESSES  IN  SLABS. 


67 


range  of  concretes  and  the  range  in  the  amounts  of  reinforcement  used  in 
the  beams  represented,  the  agreement  between  the  test  results  and  the 
empirical  equations  seems  good. 

In  Fig.  31  and  32  the  slopes  of  the  lines  which  represent  the  stages  of 
the  test  in  which  the  concrete  had  not  cracked  were  approximately  inversely 
proportional  to  the  values  of  the  modulus  of  elasticity  of  the  concrete. 
The  slopes  (1.04/8)  of  all  the  lines  for  the  cinder  concrete  beams  were 
just  twice  as  great  as  the  slopes  (0.52/8)  for  the  corresponding  beams  of 
stone  or  gravel  concrete.  The  values  of  n  (the  ratio  of  the  modulus  of 


10000 


20000  30000 

5f/TS55,  /£,  A 


4000O         5QOOO 


0 


FIG.  34. — RELATION  BETWEEN  OBSERVED  AND  COMPUTED  TENSILE  STRESSES 
IN  REINFORCEMENT  OF  BEAMS  TESTED  AT  ST.  Louis. 

Test    results    for    beams    of   cinder    concrete    compared    with    results    from    derived 
equation. 

elasticity  of  the  steel  to  that  of  the  concrete)  were,  on  the  average,  2.33 
times  as  great  for  the  cinder  concrete  as  for  the  stone  and  the  gravel 
concretes.  Assuming  that  the  value  of  the  slope  may  be  taken  as  propor- 
tional to  the  value  of  n  an  equation  is  found  which  gives  values  closelj 
approximating  the  test  results  when  the  proper  values  of  n  are  used  in  the 
equation.  The  equation  is 

f\"7  mf 

(5) 


68  MOMENTS  AND  STRESSES  IN  SLABS. 

For  all  the  beams  of  stone  or  gravel  concrete  reported  in  Technologic 
Paper  No.  2  the  average  of  the  unit-deformations  in  the  reinforcement  at 
the  time  that  the  first  crack  was  observed  is  0.000113  and  this  corresponds 
to  a  stress  of  3390  Ib.  per  sq.  in.  For  all  the  cinder  concrete  beams  reported 
in  that  paper,  the  average  unit-deformation  at  the  occurrence  of  the  first 
crack  was  0.000179.  This  deformation  corresponds  to  a  stress  of  5370  Ib. 
per  sq.  in.  The  intersections  of  the  two  straight  portions  of  the  diagrams 
of  Fig.  31  for  the  stone  and  gravel  concretes,  lie  at  an  observed  tensile 
stress  of  about  3200  Ib.  per  sq.  in.  In  Fig.  32  the  intersections  for  the 
cinder  concrete  lie  at  a  tensile  stress  of  6260  Ib.  per  sq.  in.  These  values 
are  seen  to  correspond  quite  closely  to  the  stresses  at  which  the  first  cracks 
were  discovered. 

For  the  stage  of  the  test  above  the  cracking  of  the  concrete  the  only 
difference  between  the  equation  which  represents  the  relation  between  the 
observed  and  the  computed  stresses  for  the  stone  and  gravel  concrete 
(  equation  (  3  )  )  and  the  corresponding  equation  for  the  cinder  concrete 
beams  (equation  (4)  )  is  that  in  the  former  there  is  an  additive  term 
(  —  3600  Ib.  per  sq.  in.)  which  is  lacking  in  the  equation  for  the  cinder 
concrete  beams.  It  may  seem  unexpected  that  such  a  term  as  this  should  be 
present,  but  that  the  difference  expressed  by  the  term  is  present  is  made 
entirely  clear  by  attempting  to  fit  the  equation  for.  the  cinder  concrete  to 
the  results  for  the  stone  and  the  gravel  concretes. 

The  intensity  of  the  bond  stresses  between  cracks  will  be  affected  by 
variations  in  the  modulus  of  elasticity  of  the  concrete,  and  it  may  be  per- 
missible to  assume  that  the  variation  in  the  additive  term  in  equation  (3) 
is  proportional  to  the  variation  in  the  value  of  n  (the  ratio  of  the  modulus 
of  elasticity  of  the  steel  to  that  of  the  concrete).  With  this  as  an  assump- 
tion a  more  general  equation  which,  for  the  beams  under  consideration, 
represents  quite  accurately  the  relation  between  the  observed  tensile  stress 
and  the  computed  stress  after  the  concrete  had  cracked  is 


17.  OBSERVED  TENSILE  STRESS  AT  MAXIMUM  LOAD.  It  is  desirable  to 
determine  the  relation  between  the  maximum  loads  which  the  beams  car- 
ried and  the  stress  in  the  steel  which  corresponds  to  the  observed  deforma- 
tion (here  termed  observed  stress)  at  those  loads.  On  account  of  the 
possibility  that  the  steel  had  been  stressed  beyond  the  proportional  limit 
before  reaching  the  maximum  load  it  is  not  feasible  to  determine  the  stress 
at  the  maximum  load  directly  from  the  measured  deformation.  In  order  to 
determine  the  desired  relation,  the  straight  lines  BC  of  Fig.  29  were  pro- 
duced to  the  maximum  load,  and  the  unit-deformation  given  by  this  line  at 
the  maximum  load  was  used  to  determine  the  stress  at  that  load.  In  this 
way  the  ratios,  q,  of  the  stress  at  maximum  load  to  the  yield  point  were 
determined  and  are  given  in  Fig.  35.  The  equation  which  expresses  the 
average  relation  between  q  and  the  ratio  of  reinforcement,  p,  is 

q-0.82  +  7p.  (7) 


MOMENTS  AND  STRESSES  IN  SLABS. 


69 


The  yield-point  stress  used  in  these  computations  was  40,000  Ib.  per  sq.  in. 
The  observed  tensile  stress  at  the  maximum  load  was  generally  slightly 
less  than  the  yield  point.  It  is  possible  that  the  stress  at  a  crack  was 
enough  greater  than  the  stress  found  from  the  deformations  over  the 
entire  gage  length  to  bring  the  stress  at  the  maximum  load  up  to  the  yield 
point.  This  possibility  is  further  indicated  by  the  fact,  which  is  brought 
out  in  equation  (7),  that  the  observed  stress  approached  the  yield  point 
more  closely  for  the  beams  with  a  large  percentage  of  reinforcement  than 
for  the  beams  with  a  small  percentage.  The  result  expressed  in  equation 
(7)  should  not  be  unexpected  since  the  bond  stresses  between  cracks  would 
have  more  influence  in  reducing  the  total  deformations  in  beams  in  which 
the  amount  of  reinforcement  is  small  than  in  those  in  which  it  is  large. 


0      .002   .004    .006  .006     .O/O    .012    .014     .O/6     .0/8  -020     .022 


FIG.   35.— RELATION   BETWEEN  RATIO   OF   REINFORCEMENT  AND   RATIO  OF 
OBSERVED  STRESS  AT  MAXIMUM  LOAD  TO  YIELD-POINT  STRESS. 

Equation  (7)  has  been  introduced  into  the  diagrams  of  Fig.  31  and  32 
to  show  the  observed  and  the  computed  stresses  at  which  tension  failure  in 
the  reinforcement  is  likely  to  occur.  In  making  this  application  of  the 
equation  the  yield  point  was  assumed  to  be  40,000  Ib.  per  sq.  in.,  approxi- 
mately the  average  value  found  in  the  tests  of  the  coupons  taken  from  the 
beams.  No  test  data  were  available  from  which  to  show  the  relation 
between  the  stress  at  maximum  load  and  the  yield-point  stress  for  higher 
or  lower  yield  points.  However,  by  assuming  that  for  small  differences  in 
yield  point  the  loads  carried  would  be  proportional  to  the  yield-point  stress, 
the  dotted  curves  for  yield  points  of  38,000  and  42,000  Ib.  per  sq.  in.  were 
obtained.  The  error  of  these  estimates  becomes  large  for  the  beams  with 
small  amounts  of  reinforcement,  hence  these  additional  curves  were  not 
carried  beyond  the  values  for  one-half  per  cent  of  reinforcement. 

18.  FACTOR  OF  SAFETY  AGAINST  TENSION  FAILURE.  The  factor  of  safety 
for  a  structure  may  be  denned  as  the  ratio  found  by  dividing  the  working 


70 


MOMENTS  AND  STRESSES  IN  SLABS. 


load  for  the  structure  into  the  load  which  will  cause  failure  of  the  structure. 
There  may  be  differences  of  opinion  as  to  how  the  load  which  is  to  be  used 
for  determining  the  factor  of  safety  should  be  applied.  For  these  tests 
there  was  only  one  possible  load  which  could  be  considered,  the  load  which 
was  built  up  by  uniform  increments  until  failure  was  brought  about,  the 
whole  test  requiring  not  more  than  a  few  hours.  For  the  purpose  of  elimi- 
nating from  the  study  of  the  factor  of  safety  the  effect  of  slight  variations 
in  the  yield  point  of  the  steel,  the  maximum  loads  given  in  Technologic 
Paper  No.  2  were  corrected  to  give  a  load  which  presumably  would  have 
caused  failure  if  the  yield  point  of  the  steel  had  been  40,000  Ib.  per  sq.  in. 
The  maximum  loads  reported  were  increased  or  decreased  by  an  amount 
which  was  proportional  to  the  difference  between  the  yield-point  stress  and 
40,000  Ib.  per  sq.  in.  To  these  corrected  maximum  loads  were  added  the 
weights  of  the  beams,  and  the  resulting  loads  were  used  in  the  computation 
of  the  factor  of  safety.  The  working  load  was  taken  as  the  load  which 
gives  a  computed  tensile  stress  of  16,000  Ib.  per  sq.  in.  In  these  computa- 

TABLE  IX. — FACTORS  OF  SAFETY  AGAINST  TENSION  FAILURE. 


Kind  of  Concrete. 

Ratio  of  Reinforcement. 

Average 
Factor  of 
Safety. 

0.0049 

0.0074 

2.83 
2.92 
2.62 
2.85 

2.80 

0.0098 

0.0133 

0.0159 

0.0186 

0.0212 

2.53 
2.78 
2.70 
2.36 

2.59 

3.09 
3.25 
2.96 
2.96 

3.06 

2.75 
2.88 
2.65 
2.74 

2.76 

2.69 
2.95 
2.64 
2.68 

2.74 

2.67 
2.88 
2.48 
2.56 

2.65 

2.52 
2.77 
2.51 
2.63 

2.61 

2.72 
2.92 
2.65 
2.68 

2.74 

Gravel 

Limestone  

Cinder  

Average  

tions  the  value  of  ;  (the  ratio  of  the  moment  arm  to  the  depth  d)  was 
taken  as  0.875.  The  factors  of  safety  found  in  this  way  are  shown  as 
plotted  points  in  Fig.  36.  Each  point  represents  the  average  for  three 
beams.  The  values  from  which  these  points  were  plotted  are  given  in 
Table  IX. 

From  the  definition  of  the  factor  of  safety  it  will  be  seen  that  the  ratio 
of  the  computed  stress  at  the  maximum  load  to  the  computed  stress  at  the 
working  load  will  be  the  factor  of  safety.  From  this  relation  another 
method  of  determining  the  factor  of  safety  is  afforded,  since  Fig.  31  and  32 
show,  more  or  less  exactly,  the  values  of  the  computed  stress  in  tension  at 
the  maximum  load.  These  -stresses  divided  by  16,000  Ib.  per  sq.  in.  give  the 
values  of  factor  of  safety  shown  by  the  smooth  curves  in  Fig.  36.  The 
irregularities  have  been  smoothed  out  by  the  fact  that  the  values  shown 
by  these  curves  represent  the  intersections  of  mathematical  curves.  The 
dotted  portions  of  the  two  curves  express  the  indications  of  Fig.  31  and  32 
as  to  what  the  factor  of  safety  would  be  for  beams  which  have  smaller 
percentages  of  reinforcement  than  any  of  the  beams  which  form  the  basis 
of  this  study. 


MOMENTS  AND  STRESSES  IN  SLABS. 


71 


The  agreement  between  the  plotted  points  and  the  smooth  curves  is  as 
good  as  will  generally  be  found  from  methods  which  are  as  nearly  inde- 
pendent as  were  these.  Both  sets  of  results  are  based  upon  the  same 
reasoning,  but  they  are  reached  by  different  methods  of  using  the  test  data. 

The  indication  of  the  smooth  curves  is  that  the  factor  of  safety  for  the 
cinder  concrete  was  less  than  that  of  the  other  concretes,  but  the  plotted 
values  do  not  bear  out  this  conclusion.  It  cannot  be  said,  however,  that  the 
arrangement  of  the  points  is  independent  of  the  kind  of  concrete.  If  an 
attempt  were  made  to  draw  a  curve  which  fits  the  values  shown  for  each 
kind  of  concrete  the  curve  for  the  gravel  and  that  for  the  limestone  con- 
crete would  be  the  highest  and  the  lowest  respectively,  while  the  curves 
for  the  granite  concrete  and  the  cinder  concrete  would  be  intermediate  and 
would  practically  coincide.  This  is  somewhat  surprising  since  both  the 


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FIG.  36. — FACTOR  OF  SAFETY  AGAINST  TENSION  FAILURE  OF  BEAMS  TESTED 

AT  ST.  Louis. 

compressive  strength  and  the  modulus  of  elasticity  were  highest  for  the 
granite  concrete  and  lowest  for  the  cinder  concrete.  This  would  indicate 
that  it  is  some  other  property  of  the  beams  than  the  strength  of  the  con- 
crete which  determined  the  differences  in  the  factor  of  safety  developed. 

It  is  of  interest  that  a  tendency  toward  a  higher  factor  of  safety  for 
the  beams  with  a  small  amount  of  reinforcement  than  for  the  beams  with 
large  amounts  of  reinforcement  appears  in  this  figure.  Apparently,  as  a 
criterion  of  the  load-carrying  capacity  of  the  beams  throughout  the  range 
of  the  tests  the  yield  point  of  the  steel  is  much  more  nearly  correct  than  is 
the  ultimate  strength  of  the  steel.  It  is  quite  clear  that  this  figure  gives  no 
basis  for  claiming  a  factor  of  safety  of  four  for  a  beam  having  the  usual 
amount  (say  0.7  per  cent)  of  reinforcement  of  structural  grade,  if  the 
working  stress  in  tension  is  16,000  Ib.  per  sq.  in.  An  estimate  of  the 
factor  of  safety  of  these  beams,  based  upon  the  ultimate  strength  of  the 
steel,  would  give  from  3.25  to  4.0.  These  and  many  other  tests  have  made 


72  MOMENTS  AND  STRESSES  IN  SLABS. 

it  clear  that  such  a  basis  of  estimating  the  factor  of  safety  is  wrong,  and 
yet  occasionally  a  claim  of  such  a  factor  of  safety,  arrived  at  in  this 
manner,  is  made. 

19.  SUMMARY,  (a)  It  was  found  that  the  load-strain  diagrams  for  the 
beams  could  be  represented  quite  closely  by  two  straight  lines  which  inter- 
sect at  a  point  which  corresponds  to  the  strain,  in  the  tension  side  of  the 
beam,  at  which  the  concrete  cracked  in  tension.  Through  a  study  of  the 
average  slopes,  and  average  intercepts  of  these  lines,  it  was  found  to  be 
possible  to  state  equations  which  give,  with  a  considerable  degree  of 
accuracy,  the  relation  between  the  observed  and  the  computed  stresses  in 
the  reinforcement  of  the  beams. 

(b)  The  relation  between  the  observed  and  the  computed  stresses  in 
the  reinforcement  for  the  beams  studied  was  found  to  be  affected  by  the 
variation  in  the  quality  of  the  concrete,  the  amount  of  reinforcement,  and 
the  intensity  of  the  computed  stress. 

(c)  For   stages   of  the  test  below  the  cracking  of  the  concrete  the 
rate  of  increase  of  the  tensile  deformation  was  affected  in  an  important 
degree  by  the  quality  of  the  concrete,  while  the  effect  of  the  amount  of 
reinforcement  on  the  rate  of  increase  of  tensile  deformation  was  almost 
negligible. 

The  rate  of  increase  in  the  tensile  deformation  in  the  reinforcement 
at  this  stage  of  the  test  was  found  to  be  approximately  proportional  to  the 
reciprocal  of  the  modulus  of  elasticity  of  the  concrete  in  compression  as 
determined  by  tests  of  control  cylinders. 

(d)  For  stages  of  the  test  above  the  cracking  of  the  concrete  the  rate 
of  increase  of  the  tensile  deformation  was  affected  in  an  important  degree 
by  the  amount  of  reinforcement,  while  the  effect  of  the  quality  of  the  con- 
crete on  the  rate  of  increase  in  deformation  was  entirely  negligible. 

The  total  amount  of  deformation,  however,  was  found  to  be  greater  for 
the  beams  of  cinder  concrete  than  for  the  beams  having  a  greater  com- 
prcssive  strength  and  modulus  of  elasticity.  The  difference  in  amount  of 
the  deformation  for  the  different  concretes  was  constant  for  all  percentages 
of  reinforcement  and  for  all  stages  of  the  test  between  the  cracking  of  the 
concrete  and  the  reaching  of  the  maximum  load,  as  far  as  the  data  of  the 
tests  give  a  basis  for  judgment  on  this  subject. 

(e)  The  observed  tensile  stress  in  the  reinforcement  was  less  than  the 
computed  stress  for  all  loads  up  to  and  including  the  maximum  load.     The 
difference  was  greater  both  proportionally  and  quantitatively  for  the  beams 
with   small  percentages  of  reinforcement  than  for  beams  with  large  per- 
centages.    This  was  true  for  all  loads.     Correspondingly  for  all  percentages 
of  reinforcement  the  difference  was  greater  for  low  loads  than  it  was  for 
high  loads. 

(f)  The  indications  were  that  with  a  reinforcement  of  not  less  than 
0.2  per  cent  the  strength  of  the  reinforced  beam  would  be  the  same  as  the 
strength   of   an   unreinforced   beam.      This   holds   for   the   cinder   concrete 
beams  as  well  as  for  those  made  with  concrete  of  a  higher  compressive 


MOMENTS  AND  STBESSES  IN  SLABS.  73 

strength  and  higher  modulus  of  elasticity.  Since  no  beams  with  less  than 
0.49  per  cent  of  reinforcement  were  tested  this  observation  must  be  taken 
as  an  indication  and  not  as  a  fact  established  for  the  beams  studied. 

(g)  The  observed  stress  in  the  reinforcement  at  the  maximum  load 
was  found  to  be  less  than  the  yield  point  for  all  the  beams  studied.  It  was 
found,  however,  that  the  ratio  of  the  stress  at  maximum  load  to  the  yield 
point  was  greater  for  the  beams  with  large  percentages  of  reinforcement 
than  for  the  beams  with  small  percentages. 

(h)  The  average  factor  of  safety  (see  Art.  18  for  definition)  was  found 
to  be  10  per  cent  greater  than  the  ratio  of  the  yield  point  of  the  reinforce- 
ment to  the  working  stress  in  tension  which  was  used  in  determining  the 
working  load.  When  averages  for  all  the  concretes  are  considered  a  con- 
sistent decrease  in  the  factor  of  safety  with  increase  in  percentage  of  rein- 
forcement was  found.  The  factor  of  safety  was  18  per  cent  greater  for 
beams  with  0.49  per  cent  of  reinforcement  than  for  beams  with  2.12  per 
cent  of  reinforcement. 


IV.— TESTS   OF   SLABS   SUPPORTED   ON   FOUR   SIDES. 
BY  H.  M.  WESTERGAABD. 

20.  TESTS  OF  SLABS  SUPPORTED  ON  FOUR  SIDES.  Information  as  to  the 
strength  of  slabs  supported  on  four  sides  was  obtained  by  a  series  of  tests 
made  during  the  years  1911  to  1914  in  Stuttgart  in  Germany  under  the 
direction  of  Bach  and  Graf,*  and  by  a  test  made  in  1920  at  Waynesburg, 
Ohio,  for  J.  J.  Whitacre,  under  the  direction  of  W.  A.  Slater. 

In  Bach's  and  Graf's  test  52  slabs,  simply  supported  along  the  edges, 
and  35  control  strips,  supported  as  beams,  were  loaded  to  failure.  The 
strength  of  the  slabs  was  to  be  compared  with  the  strength  of  the  strips.  A 
record  of  the  progress  of  the  test  was  obtained  by  measuring  the  deflections 
at  a  number  of  points  and  the  slopes  at  the  centers  of  the  edges,  and  by 
observing  the  development  of  cracks.  Fig.  37  and  Fig.  38  show  typical 
examples  of  the  record  made  of  the  cracks;  the  numbers  indicate  the  loads 
in  metric  tons  at  which  the  particular  cracks  appeared. 

Two  mixtures  of  concrete  were  used,  with  the  following  properties: 

Type  A  B 

Mixture   ". 1:2:3  1:3:4 

Per  cent  of  water   9.2  9.7 

Prism    strength    in    compression    after    44-48 

days,  Ib.  per  sq.  in 2,290  1,835 

Initial   modulus   of   elasticity   in   compression, 

Ib.  per  sq.  in 4,050,000  3,400,000 

The  1:  3:  4  mixture  was  used  in  three  slabs,  which  were  designed  to  fail 


*  Reported    by    Bach    and    Graf    in    Deutschcr    Ausschuss    fur    Eisenbeton,    v.    30, 
1915.     See  the  Bibliography  in  Appendix  C. 


74 


MOMENTS  AND  STRESSES  IN  SLABS. 


in  compression  (slabs  g  in  Table  II)  and  in  the  corresponding  six  control 
strips  ( 26  and  27  in  Table  X ) .  The  1:2:3  mixture  was  used  in  all  the 
other  slabs  and  strips.  At  the  time  of  the  test  the  age  of  the  specimens 
was  from  40  to  54  days.  The  yield  point  of  the  steel  was  from  49,600  Ib. 
per  sq.  in.  to  75,200  Ib.  per  sq.  in.  The  slabs  were  two-way  reinforced,  with 
the  bars  parallel  either  to  the  sides  or  to  the  diagonals. 


FIG.  37. — TOP  OF  SQUARE  SLAB  OF  200  CM.  SPAX,  TESTED  BY  BACH  AND  GRAF. 

Table  X  and  Fig.  39  show  certain  results  of  the  tests  of  the  control 
strips.  In  order  to  imitate  the  conditions  of  the  two-way  reinforced  slabs 
most  of  the  strips  were  built  with  transverse  bars  either  above  or  below 
the  longitudinal  reinforcement  (as  indicated  in  one  of  the  columns  in 
Table  X).  The  transverse  bars  were  found  to  hasten  the  development  of 


MOMENTS  AND  STRESSES  IN  SLABS. 


75 


the  first  crack,  but  the  table  shows  that  these  bars  have  only  little  influence 
on  the  ultimate  strength  of  the  strips.  The  strips  of  typos  18  to  25  failed 
by  tension  in  the  steel.  The  table  gives  the  modulus  of  rupture  of  the 
steel,  that  is,  the  steel  stress  computed  by  the  ordinary  theory  of  reinforced- 
concrete  with  n  =  15,  for  the  observed  maximum  load  (with  the  dead 


FIG.  38. — BOTTOM  OF  SQUARE  SLAB  OF  200  CM.  SPAN,  TESTED  BY  BACH  AND 

GRAF. 


weight  taken  into  consideration).  In  the  strips  that  failed  in  tension  the 
ratio  of  the  modulus  of  rupture  of  the  steel  to  the  yield  point  of  the  steel 
was  found  to  be  approximately  /  If  -  1.26  (1  —  lOp),  where  p  is  the 
ratio  of  steel.  The  strips  of  types  26  and  27  failed  in  compression  at  an 
average  modulus  of  rupture  of  the  concrete  of  3490  Ib.  per  sq.  in.  (computed 
with  n  —  15),  that  is,  1.90  times  the  prism  strength. 


76 


MOMENTS  AND  STRESSES  IN  SLABS. 


Table  XI*  and  Fig.  40  show  some  of  the  results  obtained  from  the  tests 
of  40  slabs  out  of  the  52  which  were  tested.  The  surface  of  each  of  the 
square  slabs,  a  to  g  of  the  rectangular  slabs  h,  and  of  the  rectangular  slabs 
i,  was  divided  into  50  cm.  squares;  that  is,  into  16,  24,  and  32  squares,  respect- 
ively; equal  loads  were  applied  at  the  centers  of  these  square's,  each  load 


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*  Table  XI  is  modeled  after  similar  tables  used  by  E.  Suenson  (Ingenioeren, 
1916,  p.  541)  and  by  N.  T.  Nielsen  Hngenioeren,  1920.  p.  724),  in  the  studies  of  the 
same  experimental  material.  Table  XI  was  computed  from  the  original  data  reported 
by  Bach  and  Graf;  the  results  computed  here  agree  approximately  with  those  found 
by  Suenson  and  by  Nielsen.  Suenson  compared  the  experimental  coefficients  of 
moment  in  rectangular  slabs  with  those  found  by  an  approximate  theory.  Nielsen 
computed  the  stresses  by  using  coefficients  which  he  derived  by  the  method  of  difference 
equations. 


MOMENTS  AND  STRESSES  IN  SLABS. 


77 


distributed  within  a  circle  9  cm.  in  diameter.  Thus,  the  loads  on  each  of 
these  slabs  were  nearly  uniformly  distributed.  Of  the  remaining  twelve 
slabs,  not  included  in  Table  XI,  two  were  loaded  by  a  concentrated  load  at 
the  center,  seven  by  eight  concentrated  loads  near  the  center,  while  three 
slabs  were  double  panels,  continuous  over  a  transverse  beam,  and  carrying 
nearly  uniformly  distributed  loads. 

If  a  square  slab,  simply  supported  on  four  sides,  is  loaded  uniformly 
by  the  total  load  W,  the  average  moment  across  the  diagonal  becomes 
1/24  W  =  0.041 7 W.  If  the  total  load  W  is  divided  into  16  concentrated 
equal  loads  applied  at  the  centers  of  16  squares  into  which  the  slab  is 
divided,  the  average  moment  across  the  diagonal  becomes  3/64  W  = 
0.0469  W.  When  these  16  loads,  instead  of  being  concentrated,  are  distrib- 


/-•«£; 
1.35 
^,.30 

\I.ZO 
l./S 
rin 

o 

no  transverse  bars, 
transverse  bars  above  long/tud/na/  steel, 
transverse  barj  below  longitudinal  steel. 
Each  point  represents  the  average  for 
three  or  four  strips 
,                                                           i 



*^-^ 

^-i 

o 

~—  -^^ 

/^=f.?6(> 

-**) 



r^C, 

^ 

^**^**x 

^^^. 

5T~-^ 

.__.     .OOQ    .OO9    .OlO 

Fatio  of  steel,  p 

FIG.  39. — RATIOS  OF  MODULUS  OF  RUPTURE    (COMPUTED  ULTIMATE  STEEL 
STRESS,  /,  \   TO  YIELD  POINT     /•     ix  CONTROL  STRIPS  TESTED  BY  BACH 

Jbsl  >    Jyi 

AND  GRAF. 

uted  uniformly  within  small  circles,  drawn  at  the  centers  of  the  squares, 
with  a  diameter  equal  to  9/200  of  the  span,  then  the  average  moment  per 
unit-width  across  the  diagonal,  as  may  be  verified  by  a  simple  statical 
computation,  becomes  0.0460W,*  that  is,  1.104  times  the  moment  due  to  a 
uniform  load.  An  equivalent  uniform  load  was  derived,  therefore,  in  Table 
XI,  for  the  square  slabs,  by  multiplying  the  load  applied  at  16  points  by 
1.104  and  by  adding  the  dead  loads,  1200  kg  and  800  kg  for  the  12  cm  and 
8  cm  slabs,  respectively  (that  is,  the  dead  weights  within  the  supporting 
edges).  In  the  rectangular  slabs  the  section  of  maximum  stress  is  not 
along  the  diagonal;  for  these  slabs  the  equivalent  uniform  load  was  com- 
puted as  the  sum  of  the  applied  load  and  the  dead  load.  The  following 
coefficients  of  moments  were  used  in  Table  XI:  0.0417,  the  coefficient  of  the 
average  moment  across  the  diagonal,  was  used  for  all  the  square  slabs;  in 


•  E.    Suenson,    in    Ing-enioeren,    1916.    p.    538,    states    this    moment    as   - 


0.0459  1C. 


21. S    W    = 


78 


MOMENTS  AND  STRESSES  IN  SLABS. 


addition,  the  coefficients  0.0369  and  0.0463,  applying  to  the  center  and  to 
the  corner,  respectively, — see  Fig.  3 (a)  in  Art.  7, — were  used  for  the  square 
slabs  with  non-uniform  spacing  of  the  steel  (slabs  d^  and  d2)  ;  finally, 
0.0733  and  0.0964,  coefficients  of  moment  per  unit-width,  in  the  short  span 
at  the  center,  taken  from  Fig.  3 (a)  in  Art.  7,  were  used  for  the  rectangular 
slabs  h  and  i.  Since  the  bending  moments  computed  for  the  square  slabs 
are  moments  across  the  diagonal,  the  section  modulus  pjd?  per  unit-width 
may  be  taken  as  the  average  for  the  two  layers  of  steel  if  the  steel  is  par- 
allel to  the  sides;  in  the  slabs,  /„  /a,  and  g,  with  reinforcement  parallel  to 


160 
>55 
f.& 

>45 

!+} 

f] 

^c/77.  slabs 
*>  cm.  slabs 
';  square  slabs,  ZOOcmxZooc 
•ecianqular  slabs,  3X)cm*30C 
octangular  slabs,  ?OOcm.x40L 

- 

o 

+    i 

/- 

Keck. 

'/TJfcife 

rsla 

tf 

tor 

m. 
vn. 

i"*4 

i;r 

fc/7Z 

* 

s* 

^ 

on-ur 

iforn 

?^a7^ 

;inq 

1.20 

H5 
HO 

i'Ti 

r 

~5arr 

lesp 

acin^ 

T  inh 

wo  tc, 

lyers 

J 

+GZ 

C/05< 

?rsp<. 

.   \ 
Tcinq  in  ut 

oper 

layer 

^ 

----. 

^4 

k^ 

-       -^ 

^ 

—L»- 

* 

G 

0- 

<* 

>^. 

1  —  , 

^^ 

-Con 

"~>«^ 

troh 

tripste-^ 
(y 

126(1 

-lOp) 

\5; 

eel  parallel 
to  diagonals 

-~~^ 

£ 

ClQSt. 

'.rspc. 

Tang 

in  Uf. 

iperk 

•jyer 

1.00 

L 

'      5 

fee/f. 

•>arali 

el  -to 

d/agc 

ma/5 

*9 

0     .001   .002  .003  .004   .005   006   007    .008   .009   .010    .Oil    .OIZ     .013    0/4    Qi 

of  s  fee/,  p 
FIG.  40. — RATIOS  OF  MODULUS  OF  RUPTURE     f      TO  YIELD  POINT 

>      /S)  > 

SLABS  SUPPORTED  ox  FOUR  SIDES,  TESTED  BY  BACH  AND  GRAF. 

d,  AND  rfj  REPRESENT  AVERAGES  OF  Two  TESTS  EACH,  OTHER  POINTS 

AVERAGES  OF  THREE  TESTS  EACH. 


/      IN 

JV> 

POINTS 


the  diagonal,  the  section  moduli  in  the  two  directions  are  practically  equal 
and  the  average  value  may  be  used  again.  In  the  rectangular  slabs,  h  and  i, 
the  section  modulus  by  which  the  stresses  are  computed  may  be  taken  as 
that  defined  by  the  bottom  layer,  which  is  in  the  direction  of  the  short  span. 
As  an  example  of  the  computations  in  Table  XI  the  derivation  of  the 
stress  f  in  the  slabs  Oj  may  be  shown : 


M          0.0417   •  45700kg     u  22 


Ib.cm2 


=  70000 

in2 


pjd*   *  0.387cm2  "   kg.in2 

The   stress    f     is   the   steel   stress,   computed  by  the  ordinary   theory  of 


MOMENTS  AND  STRESSES  IN  SLABS. 


79 


reinforced-concrete  with  n  =  15,  developed  under   the  observed   maximum 
load;    that  is,  t    is  tlie  modulus  of  rupture  in  bending. 

The  stresses   f     developed  in  the  slab  may  be  judged  by  comparison 
with  the  yield  point  /•     of  the  steel  and  the  modulus  of  rupture    f     (level- 


oped  in  the  strips.    Such  comparisons  are  made  in  the  last  three  columns  in 
Table  XI.     The  ratios  fjf    are  represented  graphically  in  Fig.  40. 

The  slabs  e  were  designed  to  fail  in  compression.     The  stresses  devel- 
oped were:    tension,  39,600  Ib.  per  sq.  in.;    compression,  3430  Ib.  per  sq.  in.. 


80  MOMENTS  AND  STRESSES  IN  SLABS. 

which  is  0.983  times  the  corresponding  stress  developed  in  the  strips,  and 
1.87  times  the  prism  strength  of  the  concrete  in  compression. 

Certain  conclusions  may  be  drawn  from  Table  XI  and  from  Fig.  40: 

(a)  The  slabs  show,  on  the  whole,  the  same  decrease  of  modulus  of 
rupture  with  an  increasing  ratio  of  steel  as  did  the  strips  which  were 
tested  as  beams. 

(b)  The  thinner  slabs  develop,  on  the  whole,  greater  moduli  of  rupture 
than  the  thicker  slabs  with  the  same  span  and  reinforcement.     This  result 
may  be  explained  by  the  dish  action  which  occurs  when  the  deflections  have 
become  appreciable  compared  with  the  thickness  of  the  slab.     The  slabs 
a,  and  a,,  for  example,  which  were  12  cm  and  8  cm  thick,  respectively, 
deflected  about  6  cm  at  the  center  at  the  maximum  load.     By  the  double 
curvature  of  a  slab  the  vertical  sections  resisting  the  bending  moments 
assume  an  arc-shape  instead  of  the  original  rectangular  shape,  and  thus  the 
section  modulus  is  increased.    At  a  given  deflection,  this  effect  is  compara- 
tively greater  in  a  thin  slab  than  in  a  thick  slab.     The  dish  action  of  the 
thin  slab  may  be  interpreted  as  a  reversed  dome  action,  in  which  the  central 
area  is  essentially  in  tension,  while  the  outer  area  is  essentially  in  com- 
pression.     The    additional   tensions   and    compressions   explain    the   added 
carrying  capacity,  beyond  what  may  be  expected  on  the  basis  of  the  coeffi- 
cients of  moment  which  were  derived   for  the  medium-thick   stiff  homo- 
geneous elastic  plates. 

(c)  The  design  with  closer  spacing  of  the  bars  in  the  upper  than  in 
the  lower  layer  of  steel,  so  as  to  make  the  section  moduli  equal  for  the  two 
layers,  does  not  appear  to  be  advantageous. 

(d)  Reinforcement  parallel  to  the  diagonals  appears  to  be  less  effective 
than  reinforcement  parallel  to  the  sides.    If  the  corners  had  been  prevented 
from    bending    up    by    anchoring, — the    corners    were    observed    to    deflect 
slightly  upward, — and  if  the  steel  along  the  diagonal  had  been  bent  up  so 
as  to  reinforce  against  negative  moments  at  the  corner,  greater  strength 
might  possibly  have  been  developed  with  the  same  amount  of  steel. 

(e)  The  slab  has  an  ability  to  redistribute  the  stresses  as  the  deflec- 
tions increase,  as  the  steel  stresses  approach  or  reach  the  yield  point,  and 
as  cracks  develop.     By  the  redistribution  the  large  stresses  become  smaller 
and  the  small  stresses  larger  than  would  be  predicted  according  to  the  dis- 
tribution in  the  homogeneous  elastic  slabs  for  which  the  theory  in  Part  II 
was  derived.     The  phenomenon  of  redistribution  is  well  known  from  other 
fields.     For  example,  in  a  flat  steel  tension  bar  with  a  circular  hole  there 
is,  at  small  stresses,  a  relative  concentration  of  stress  at  the  edge  of  the 
hole,  but  when  the  yield  point  has  been  reached  the  stresses  may  be  prac- 
tically uniformly  distributed.     Redistribution  of  stresses  is  a  typical  gen- 
eral  feature   in   statically   indeterminate   structures   of   ductile   materials. 
Thus,  the  property  of  the  slab  as  a  highly  statically  indeterminate  structure 
becomes   important;     it   explains   additional   strength   beyond   what  might 
otherwise  be  expected.     In  the  homogeneous  elastic  square  slab  with  simple 
supports  on  four  sides  and  with  Poisson's  ratio  equal  to  zero  the  coeffi- 
cients of  moment  per  unit-width  across  the  diagonal  are,  according  to  Art. 


MOMENTS  AND  STRESSES  IN  SLABS  &L 

7:  at  the  corner,  0.0463;  at  the  center,  0.0369;  average  for  the  whole 
diagonal,  0.0417.  The  stresses  across  the  diagonal  may  be  redistributed  so 
as  to  become  nearly  uniform;  accordingly  the  average  coefficient,  0.0417, 
was  applied  to  all  the  square  slabs  in  Table  XI.  The  maximum  coefficient 
0.0463  would  have  made  the  corresponding  stresses  and  ratios  of  stresses 
in  Table  XI  and  in  Fig.  40,  1.11  times  greater  than  the  values  shown.  The 
coefficients  0.0463  would  have  led  to  a  less  close  agreement  between  the 
slab  strength  and  the  strip  strength  than  is  found  in  Table  XI  and  in  Fig.  40. 
The  redistribution  of  stresses  across  the  diagonal  may  explain  the  rather 
large  stresses  computed  for  the  corners  of  the  slabs  d^  and  dt.  These 
stresses  were  computed  by  using  the  largest  coefficient  of  moment  in  con- 
nection with  the  smallest  percentage  of  steel.  Evidently  the  stresses  have 
transferred  toward  the  center,  where  the  spacing  of  the  steel  is  closer,  and 
as  a  result  of  this  redistribution  the  steel  at  the  center  appears  to  be  more 
effective,  per  pound  weight,  in  resisting  the  utlimate  loads,  than  the  steel 
near  the  edges.  In  the  rectangular  slabs  the  rather  large  moduli  of  rupture, 
84,900  and  95,500  Ib.  per  sq.  in.,  computed  by  the  coefficients  of  moment  for 
the  short  span  at  the  center,  may  be  explained  partly  by  a  redistribution 
of  the  stresses  across  the  long  center  line,  whereby  the  actual  stresses  at  the 
center  are  reduced,  and  partly  by  a  transfer  of  stresses  from  the  short  span 
into  the  long  span.  N.  J.  Nielsen,*  by  using  moment  coefficients  found  by 
the  method  of  difference  equations,  with  Poisson's  ratio  equal  to  zero,  deter- 
mined ratios  of  slab  strength  to  strip  strength  for  the  square  slabs  loaded 
nearly  uniformly,  for  the  square  slabs  loaded  by  eight  forces  near  the 
center,  for  the  square  slabs  loaded  by  one  force  at  the  center,  and  for  the 
rectangular  plates.  He  found  the  ratio  of  slab  strength  to  strip  strength  to 
be  fairly  uniform  for  all  the  slabs,  including  the  rectangular  slabs,  by 
assuming  different  values  of  the  moment  of  inertia  for  the  two  spans, 
namely,  such  values  that  the  maximum  computed  stresses  become  equal  in 
the  two  spans,  with  the  steel  in  the  two  directions  utilized  fully. 

A  comparative  study  of  computed  and  observed  deflections  of  one  of 
the  double  panels  ( two  square  panels,  continuous  over  a  transverse  beam ) , 
under  a  load  equal  to  about  one-fourth  of  the  ultimate  load,  was  made  by 
X.  J.  Nielsen,f  who  used  the  method  of  difference  equations.  By  considering 
the  plate  as  made  of  homogeneous  material  with  a  modulus  of  elasticity  of 
4,110,000  Ib.  per  sq.  in.,  and  by  taking  the  deflections  of  the  transverse 
beam  as  observed,  he  found  the  computed  and  the  observed  deflections  at 
the  centers  of  the  panels  to  be  equal,  and  he  found  the  deflection  curves 
and  contour  lines  shown  in  Fig.  41.  Near  the  transverse  beam  the  observed 
deflections  are  seen  to  be  smaller  than  the  computed  deflections.  Thi-s 
difference  may  be  due  to  the  rather  heavy  reinforcement  across  the  trans- 
verse beam. 

The  test  made  in  1920  at  Waynesburg,  Ohio,  for  Mr.  J.  J.  Whitacre, 
throws  further  light  on  the  question  of  the  redistribution  of  stresses,  as 
compared  with  the  stresses  in  homogeneous  elastic  slabs,  and  on  the  ques- 


*  Ingenioeren,  1920,  p.  724. 

tN.  J.  Nielsen,  Spaemlinger  i  Plader,  1920,  p.  74. 


82 


MOMENTS'  AND  STRESSES  IN  SLABS. 


tion  of  the  ultimate  strength  of  the  slabs.*  The  test  was  made  with  a  two- 
way  reinforoed-concrete  and  hollow  tile  Hour  slab,  0  in.  thick,  with  18 
panels.  Fig.  42  shows  the  plan  of  the  floor.  The  tiles  are  G  in.  by  12  in. 
by  12  in.,  open  at  the  ends  so  as  to  allow  the  concrete  to  How  in,  filling  up 
a  part  of  the  tile.  The  tiles  are  separated  by  4-in.  concrete  ribs  in  both 
directions.  Each  rib  is  reinforced  by  a  ^-in.  round  bar  at  the  bottom,  and, 
in  addition,  in  the  part  of  the  rib  near  the  panel  edges,  by  a  ^-in.  round 
bar  at  the  top.  The  yield  point  of  the  steel  was  54,000  Ib.  per  sq.  in. 

Large  negative  moments  were  produced  at  the  edges  by  loading  all  the 
panels  and  the  cantilevers  adjacent  to  B,  C,  and  E,  at  the  same  time. 
Results  of  this  part  of  the  test  are  shown  in  Table  XII.  The  applied  load 
on  each  panel  was  a  nearly  uniformly  distributed  load,  consisting  of  four 
piles  of  bricks  with  18-in.  aisles.  Equivalent  entirely  uniformly  distributed 
applied  loads  were  derived  by  multiplying  the  average  applied  loads 
(within  the  areas  defined  by  the  clear  spans)  by  the  factors  0.91,  0.92,  and 
0.93  for  the  square,  medium  long,  and  longest  panels,  respectively;  these 


FIG.  41. — COMPARATIVE  STUDY  BY  N.  J.  NIELSEN  OF  OBSERVED  DEFLECTIONS 
(SHOWN  BY  DOTTED  LINES)  AND  COMPUTED  DEFLECTIONS  (SHOWN  BY 
FULL  LINES)  IN  A  DOUBLE  PANEL  TESTED  BY  BACH  AND  GRAF. 

factors  were  determined  by  an  approximate  theory.  The  equivalent  loads 
stated  at  the  head  of  Table  XII  are  found  by  adding  the  dead  load,  50  Ib. 
per  sq.  ft.,  to  the  equivalent  uniform  applied  load.  The  observed  stress,  /, 
at  the  center  of  the  edges,  was  considered  to  be  made  up  of  an  estimated 
dead-load  stress  of  500  Ib.  per  sq.  in.,  plus  the  increase  of  stress  due  to  the 
live  load;  this  increase  was  found  as  the  maximum  ordinate  of  a  smooth 
curve  plotted  from  strain-gage  readings  on  several  gage  lines  across  the 


*  A  detailed  report  on  this  test  has  not  yet  been  published.  Only  certain  aspects 
which  have  a  general  bearing  on  the  question  of  the  moments  and  stresses  in  slabs  are 
discussed  here. 


MOMENTS  AND  STRESSES  IN  SLABS. 


83 


particular  edge.  In  Table  XII  the  ratio  of  the  corrected  steel  stress  / 
corresponding  to  the  computed  stress  in  a  beam,  to  be  observed  stress,  /, 
has  been  determined  by  means  of  formulas  (1)  and  (3)  in  Part  III,  as 
though  the  material  were  solid  stone  or  gravel  concrete  instead  of  concrete 
and  hollow  tiles.  Since  the  reinforcing  bars  are  16  in.  apart,  the  ratio  of 
reinforcement  at  the  center  of  the  edge  is,  p  =  0.00260.  Formula  ( 1 ) , 
which  applies  at  small  stresses,  before  the  concrete  has  cracked,  gives 
then, 

/.  0.52  J_ 

7"  "  i  +  °021     17-5 
p 

and  formula    (2),  which   applies  after  the  cracking  has  begun,  gives  the 
relation 


54000 


0.222, 


by  which  the  values  were  computed  in  Table  XII.     The  computation  of  the 
-^  i 1  i 1 


f 


,     !Q 


\ 

J 

A 

B 
v  t 

C 
^  j 

D 

£ 

-<-                     v- 

F 
<,  —  .  > 

G 

.....     —  ^ 

h 
\  > 

1 

j  

J 

K 
1 

,  —           --.^ 
L 

M 

•>  ^ 

N 

0 

P 

Q 

K 

FIG.  42. — PLAN  OF  FLOOR  SLAB  TESTED  AT  WAYNESBUKG,  OHIO  ;   THICKNESS, 

6  IN. 

coefficients  of  moment  in  Table  XII  may  be  shown  by  an  example.  For  the 
edge  AB  (in  the  first  line  of  the  table)  the  observed  moment  coefficient, 
based  on  the  observed  stress,  45,000  Ib.  per  sq.  in.,  is  found  to  be 


M 


0.00260  •  0.920  •  4.7 Pin2  •  45000  .—2 
388  lb_-  152  ft2 
ft2 


Since  the  corrected  stress  is  1.422  times  the  observed  stress,  the  correspond- 
ing corrected  moment  coefficient  becomes 

1.422-  0.02735  —  0.0389. 


84  MOMENTS  AND  STRESSES  IN  SLABS. 

TABLE  XII. — COEFFICIENTS  OF  NEGATIVE  MOMENTS  IN  WAYNESBVRG  TEST. 

Test    of    reinforced  concrete   and    hollo**   fi/e  floor  slat;  IB  panels  supported 
on    girdtrs  as    shown    in   fri'ff.'-42<5trfsses  and  moments  at-centej-s  of 
edges.  All  panels  /oadfct.    fauiya/ent  trni/orm  loads  (incl  dead  load) in  /b.per  sq.ft. 

•a*Vi»  _- _  _       f_        r-t     **  >«   *  j    £_       £3      rf£?    l_f     Ay          ^/\  O  — *.    /"•   n     J       T  S\   T\      n^>t—  —     r*  fr'u     .      _'•£* 


«„«/ 

£Jf. 

0^™ 

folio 

CotfficiejiU  of  max. 

ffj* 

OltftvrJ 

Ralio 

Coefficients  of  ma*. 

tfrtts 

A 

nea.  moment    tf/wb 

ttr,» 

JL 

3'              r    / 

f 

Ot~r~J 

Cor^cffJ 

Throrj. 

f 

f 

fcer>ed 

c'trftt" 

TKrory 

1b/inl 

flf.fli) 

Ik/in1- 

/7y  8M 

-C 

fl 

flB 

45000 

uzz 

.0274 

0)89 

FIG 

Sisoo 

1.270 

.031) 

.0598 

*<\ 

& 

Bfl 

38000 

/  64) 

.02/8 

.0358 

^ 

B 

&C 

17000 

5.597 

.0098 

.03* 

BH 

3(,500 

1.70  / 

.OZIO 

.JO  3  57 

1. 

G 

Gfl 

50OOO 

I.30Z 

.0287 

.0374 

\ 

G 

GH 

19fOO 

1-589 

.0217 

.0360 

CM 

41000 

1.55? 

.02)f 

03  6Z 

^ 

H 

HG 

JJ500 

1.589 

.0211 

.0360 

HB 

48500 

1.356 

.0278 

.0371 

* 

H 

HI 

38000 

1.643 

.OUB 

.0358 

HH 

48500 

1-356 

.0178 

057Z 

s. 

fl 

tin 

44000 

1.449 

.0268 

.0388 

flG 

55000 

1.240 

O)ZZ 

.0400 

? 

/v 

fiM 

41000 

1559 

.0235 

.056% 

NH 

51500 

1.270 

0296 

.0376 

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tfO 

42500 

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ffrerage  coefficient- 

.0363 

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flreraye  coefficient: 

.0?7£ 

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29000 

2.084 

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24500 

2425 

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1744 

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.0383 

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59500 

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38000 

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2.425 

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.0674 

MOMENTS  AND  STRESSES  IN  SLABS.  85 

The  theoretical  values  of  the  coefficients  were  taken  from  the  approxi- 
mate curves  in  Fig.  8(b)  in  Art.  7.  These  values  are  somewhat  smaller 
than  the  corresponding  directly  determined  coefficients  in  Fig.  8 (a),  but 
they  may  be  assumed  to  apply  after  the  stresses  across  the  edges  have  been 
redistributed  to  some  extent.  That  a  further  shifting  of  the  stresses  from 
the  short  span  to  the  long  span  takes  place  at  increasing  deformations,  is 
indicated  by  the  fact  that  the  oblong  panels  in  Table  XII  have  corrected 
coefficients,  based  on  observations,  which  are  greater  than  the  theoretical 
values  for  the  long  span,  and  are  smaller  than  the  theoretical  values  for  the 
short  span;  but  the  sum  of  the  corrected  coefficients  for  the  long  span  and 
the  short  span  is  approximately  equal  to  the  sum  of  the  corresponding 
theoretical  coefficients.  It  is  probable  that  at  increasing  loads  there  is 
also  a  transfer  of  stress  from  the  edges  to  the  central  portion  of  each  panel; 
and  that  this  transfer  may  partly  explain  the  rather  small  values  found 
for  some  of  the  coefficients  in  Table  XII. 

In  a  later  part  of  the  test  large  loads  were  applied  in  panels  H,  J,  and 
K,  while  the  loads  on  the  surrounding  panels  were  reduced.  In  the  square 
interior  panel  H,  for  example,  the  average  applied  load  was  increased  to 
1413  Ib.  per  sq.  ft.,  giving,  by  the  computation  used  in  Table  XII,  an 
equivalent  uniform  load  of  50  +  0.91-1413  =  1336  Ib.  per  sq.  ft.  This 
value  is  probably  somewhat  too  large  because  of  the  unavoidable  arch  action 
in  the  piles  of  brick;  the  four  piles  were  joined  together  12  ft.  above  the 
slab  and  were  continued  as  one  pile  up  to  the  total  height  of  almost  22  ft. 
When  the  deflections  increase  the  resultant  pressure  transmitted  through 
each  of  the  four  piles  is  thrown  toward  the  corners  of  the  panel,  and  the 
equivalent  uniform  load  becomes  correspondingly  smaller.  Since  the 
amount  of  the  reduction  is  not  known,  the  value  just  stated,  1336  Ib.  per 
sq.  in.,  will  be  used  without  reduction  in  the  computation  of  stresses.  Since 
the  adjacent  panels  were  unloaded,  the  panel  H  may  be  considered  in  this 
computation  as  a  single  square  panel  with  the  edges  half  fixed  and  half 
simply  supported.  Accordingly,  the  average  of  the  numerical  values  of  the 
moment  coefficients  at  the  center  and  at  the  edge  in  slabs  with  simply  sup- 
ported and  with  fixed  edges  is  used,  that  is  (see  Fig.  3 (a),  Fig.  7 (a),  and 
Fig.  8 (a)). 

i  (0.0369  +  0  +0.0177  +  0.0487)  =  0.0258. 

4 

By  assuming  this  moment  coefficient,  and  by  assuming  the  same  effective 
depth  and  ratio  of  steel  as  in  the  calculation  of  Table  XII,  one  finds  the 
"computed  stress"  in  panel  H  under  the  maximum  load  equal  to 

,       0.0258  wV        0.0258  •  1336  •  152 

>•  =  -jjdr-  =  0.00260- 0.92  -4.7P  =  14600°  lb'  per  Sq'  m' 
This  stress  is  2.71  times  the  yield  point  stress  of  the  steel,  jy  =  54,000  lb. 
per  sq.  in.,  and  2.21  times  the  strip  strength,  f  bs  =66,200  lb.  per  sq.  in., 
as  determined  from  Bach's  and  Graf's  tests  by  the  line  in  Fig.  39.  In 
estimating  the  significance  of  this  result  it  should  be  noted  that  some  arch 
action  in  the  piles  of  brick  probably  made  the  applied  load  not  fully  effect- 
ive; that  the  material  is  different  from  ordinary  reinforced-concrete;  that 


86  MOMENTS  AND  STRESSES  IN  SLABS. 

the  moment  coefficients  may  have  been  reduced  by  redistribution  of  the 
stresses  across  the  center  line,  across  the  diagonal  and  across  the  edge; 
that  this  redistribution  may  have  been  aided  by  the  deflections  of  the  sup- 
porting girders;  and  that  the  deflection  at  the  center  was  so  large,  1.4  in., 
that  the  dish  action  or  reversed  dome  action  which  is  characteristic  of  thin 
slabs,  may  have  aided  in  carrying  the  load.  It  is  not  known  what  load 
would  have"  produced  failure. 

The  average  loads,  determined  by  dividing  the  total  applied  load  by 
the  panel  area,  applied  in  panels  J  and  K  at  the  same  time  and  under 
similar  conditions  without  producing  failure,  were  1184  Ib.  per  sq.  ft.  and 
920  Ib.  per  sq.  ft.,  respectively. 

V.— TESTS    OF    FLAT    SLABS. 
BY  W.  A.  SLATER. 

21.  GENERAL  DESCRIPTION.  In  the  following  pages  are  given  the 
results  of  tests  of  certain  flat  slabs.  It  has  been  necessary  to  make  the 
discussion  of  the  results  very  brief  and  only  sufficient  statement  on  each 
subject  has  been  made  to  enable  the  reader  to  interpret  the  data  given  in 
the  diagrams  and  tables. 

The  study  of  the  tests  is  based  almost  entirely  upon  tensile  stresses 
since  it  is  impossible  to  know  with  sufficient  accuracy  for  this  purpose  what 
compressive  stresses  arg  indicated  by  the  compressive  deformations  and 
because  the  amount  of  reinforcement  in  flat  slabs  is  generally  so  small  that 
the  tensile  stresses  will  almost  always  be  critical  rather  than  the  com- 
pressive stresses. 

The  results  for  most  of  the  tests  have  been  published  previously.  Those 
for  the  two  Purdue  tests,  the  Sanitary  Can  Building  test,  and  the  Shonk 
Building  test  have  not  been  published,  and  those  of  the  International  Hall 
test*  were  published  only  in  part.  Because  of  the  fact  that  the  results  of 
the  Purdue  test  had  not  been  published,  the  reinforcing  plans,  the  location 
of  gage  lines,  the  measured  deformations  and  the  deflections  are  given  in 
Appendix  B.  Further  data  are  given  in  Table  XIII. 

It  had  been  expected  to  give  the  results  for  the  Sanitary  Can  Building 
test  and  the  Shonk  Building  test  as  fully  as  for  the  Purdue  tests,  but  this 
has  not  been  possible.  The  following  statement,  together  with  the  data 
given  in  Table  XIII,  will  be  sufficient  to  give  significance  to  the  moment 
coefficients  given  in  Fig.  45  for  these  tests.  It  is  expected  that  in  a  later 
publication  of  the  Bureau  of  Standards  the  full  data  of  these  tests  will  be 
included. 

The  tests  of  the  Sanitary  Can  Building  and  the  Shonk  Building  were 
made  by  A.  R.  Lord,  of  the  Lord  Engineering  Company,  Chicago,  111.  Prof. 
W.  K.  Hatt,  of  Purdue  University,  Lafayette,  Ind.,  was  in  touch  with  these 
tests  at  the  request  of  the  Corrugated  Bar  Company.  The  report  by  Mr. 


•  Trans.  A.  S.  C.  E.,  Vol.  LXXVII,  p.  1433  (1914). 


MOMENTS  AND  STRESSES  IN  SLABS.  87 

Lord  and  that  by  Professor  Hatt  have  been  drawn  upon  for  the  data  used 
in  preparing  this  paper. 

Both  buildings  are  located  at  Maywood,  111.,  a  suburb  of  Chicago. 
The  floors  of  both  are  fiat  slabs,  having  column  capitals  5  ft.  in  diameter 
and  dropped  panels  8  ft.  square.  In  each  building  four  panels  were  loaded 
and  in  each  case  two  of  the  loaded  panels  were  wall  panels  and  the  other 
two  were  the  adjacent  interior  panels.  The  panel  length  in  the  direction 
parallel  to  the  wall  and  also  perpendicular  to  the  wall  for  the  two  interior 
panels  is  22  ft.  for  both  buildings.  For  the  wall  panel,  the  panel  length 
perpendicular  to  the  wall  is  21  ft.  3  in.  for  the  Sanitary  Can  Building  and 
20  ft.  7  in.  for  the  Shonk  Building.  The  Sanitary  Can  Building  has  24-in. 
octagonal  interior  columns  and  wall  columns  20  by  45  in.  rectangular  in 
cross  section.  The  Shonk  Building  has  22-in.  octagonal  interior  columns 
and  wall  columns  21%  in.  rectangular  in  cross  section.  The  Sanitary  Can 
Building  has  two-way  reinforcement  and  the  Shonk  Building  has  four-way 
reinforcement.  There  was  some  difference  in  distribution  of  reinforcement, 
but  in  the  two  slabs  the  total  area  provided  for  negative  moment  was 
about  the  same,  and  the  total  area  for  positive  moment  was  about  the  same. 
The  area  of  reinforcement  at  the  principal  design  sections  and  the  meas- 
ured depth  d  to  the  reinforcement  are  shown  in  Table  XIII. 

Each  floor  was  designed  for  a  live-load  of  150  Ib.  per  sq.  ft.  The 
maximum  superimposed  test  load  was  about  400  Ib.  per  sq.  ft.  The  dead 
load  brought  the  total  test  load  up  to  about  535  Ib.  per  sq.  ft.  in  each  test. 
The  highest  observed  stresses  in  the  reinforcement  were  about  24,000  Ib. 
per  sq.  in.  in  both  floors,  and  this  high  stress  occurred  in  only  a  few  places. 

The  misplacement  of  the  reinforcement  in  the  Bell  St.  Warehouse* 
probably  had  an  influence  on  the  distribution  of  resisting  moments  between 
positive  and  negative,  which  would  not  be  expected  in  slabs  of  that  type. 
At  least,  this  feature  should  be  considered  in  studying  the  results  of 
the  test. 

The  original  data  of  the* test  of  the  Western  Newspaper  Union  Building 
were  not  available,  therefore,  in  determining  the  moment  coefficients  for  this 
test,  it  was  assumed  that  the  moment  of  the  observed  tensile  stresses  at  any 
load  less  than  the  maximum  bore  the  same  ratio  to  the  resisting  moment 
reported  for  the  test  load  of  913  per  sq.  ft.f  that  the  sum  of  the  observed 
stress  and  the  estimated  dead  load  stress  for  the  load  under  consideration 
bore  to  the  .sum  of  the  observed  stress  for  the  maximum  test  load  and  the 
•estimated  dead  load  stress. 

22.  CORRECTION  OF  MOMENT  COEFFICIENTS.  In  reporting  results  of 
flat  slab  tests  it  has  been  customary  to  state  the  moment  of  the  observed 
stresses  in  the  reinforcement  as  a  proportion  of  the  product  of  the  total 
panel  load  and  the  span.  It  has  been  known  from  beam  tests  that  the 


•Pacific  N.  W.  Soc.  Civ.  Eng.,  Vol.  IS  (Jan.  and  Feb.,  1916);  Eng.  Record,  Vol. 
73,  p.  647;  Eng.  News-Record,  April  19,  1917. 

t  Bulletin  106,  Univ.  of  111.  Eng.  Exper.  Sta.,  p.  36.  Also  Proc.  A.  C.  I..  Vol. 
XIV,  p.  192  (1918). 


88 


MOMENTS  AND  STRESSES  IN  SLABS. 


total  moment  of  the  observed  stresses  is  less  than  the  applied  moment. 
The  ratio  of  the  applied  moment  to  the  moment  of  the  observed  stress  is 
equal  to  the  ratio  of  the  computed  stress  to  the  observed  stress.  This  may 
be  shown  as  follows: 

M   =  KW  I  =  Afsjd 
A/,  -  KiW  I 
from  which  3f       K      /s 


and 
where 


7v    = 


K  and  K i  are  coefficients 
KW  I  =  applied  moment 
KI  W I  =  moment  of  observed  stress 
fs  =  computed  stress 
/i  =  observed  stress 
Other  terms  have  their  usual  significance. 

J6r 


J4 


1  -  Total  Moment  f numerical 

sum  of  pas  and  neq.) 

2  -  Total  tfeqafive  Moment 

3  -  Neq.  Moment;  Col.  Head5ecf ion 

4  -  Total flssfffa  Moment 


5-Fbs.Moment;OuterSedion 
6 -Net).    -    ;Mid 
\.l-Fbs.  -  >  Inner 

i 


ZOOOO  .30000  40000  50000  60000         ^0000  30000  40000  50000  60000  70000 


Computed  Average  Tensi/e  Stress,  Ibpersq./n. 

(a)  (6) 

FIG.    43. — MOMENT   COEFFICIENT   FOR   SLAB   J;     (a)    UN  CORRECTED;     (6) 

CORRECTED. 

These  equations  show  that  for  a  beam  the  true  moment  coefficient  K^ 
may  be  determined  by  multiplying  the  moment  coefficient  of  the  observed 
stress  by  the  ratio  of  the  computed  stress  to  the  observed  stress.  This  ratio 
is  here  termed  the  moment  correction.  Although  it  is  recognized  that  the 
behavior  of  a  slab  differs  from  that  of  a  beam  it  seems  reasonable  to  assume 
that  the  relation  between  applied  moments  and  the  moment  of  the  observed 
tensile  stresses  in  the  reinforcement  should  be  the  same  for  a  slab  as  for  a 
beam  if  the  percentage  of.  reinforcement,  the  modulus  of  elasticity  of  the 
concrete,  the  depth  d  and  the  depth  of  covering  of  the  reinforcement  are  the 
same  for  the  slab  as  for  the  beam. 


MOMENTS  AND  STRESSES  IN  SLABS. 


The  beams  tested  by  the  U.  S.  Geological  Survey,  and  reported  in 
Part  III,  afford  a  basis  for  determining  the  moment  correction  for  a  wide 
range  in  the  percentage  of  reinforcement  and  the  modulus  of  elasticity  of 
the  concrete,  and  to  this  extent  the  moment  corrections  found  for  these 
beams  will  be  useful  for  estimating  from  the  observed  stress  in  the  slabs 
the  moment  applied  to  the  slabs.  The  fact  that  in  this  investigation  only 
one  depth,  d,  and  only  a  slight  variation  in  the  covering  of  the  tension 
reinforcement  were  used,  limits  the  usefulness  of  this  investigation  as 
applied  to  interpreting  test  results  of  flat  slabs,  but  no  other  series  of 
tests  is  known  which  covers  so  wide  a  range  of  conditions,  and,  notwith- 
standing these  limitations,  it  seems  reasonable  to  expect  that,  on  the  whole, 
the  application  of  the  moment  corrections  from  Fig.  31  and  Fig.  32  to  the 


16 


S 


/  -  Tola  I ' Momentfnumertcal   \ 

sum  of  pas.  and  neg)  J 

' Z -Jbfat  NsqafMsMament     \ 

~3  -  Neq.  Moment,  Caf head  Section 

4-  ~~ 


06 


/•f 


30000  40000  50000  6000O  70000      30000  40000  50000  60000   70000 


Computed  Average  Tenst/e  Stress,  /b.  per  53.  in. 
(a)  (6) 

FIG.   44. — MOMENT   COEFFICIENTS  FOR  SLAB   S;     (a)    UNCORRECTED; 

CORRECTED. 


(6) 


moment  coefficient,  determined  by  means  of  the  observed  stress  in  the  flat 

slab,  should  give  a  fair  idea  of  the  true  moment  coefficients  for  these  slabs. 

23.  MOMENT  COEFFICIENTS.     The  moment  coefficients  have  been  stated 

M 


as  values  of  the  expression 


—  in  which  M  is  the  sum  of  the 


m  0  -  a-  r 

positive  and  negative  moments  in  the  direction  of  either  side  of  the  panel, 
W  is  the  total  panel  load,  c  is  the- diameter  of  the  column  capital  and  I  is 
the  span  in  the  direction  in  which  moments  are  considered.  This  is  a  con- 
venient form  of  expression  and  it  has  been  found  possible  to  state  the 
moments  found  by  the  analysis  in  terms  of  it  witli  a  satisfactory  degree  of 


See  Art.   8. 


90 


MOMENTS  AND  STRESSES  IN  SLABS. 


The  vise  of  the  same  form  of  expression  in  stating  the  test  results 
simplifies  the  comparison  with  the  analytical  result.  In  determining  the 
value  of  M  from  the  tests  for  use  in  calculating  these  moment  coefficients 
the  equation  M  —  Afjd  was  evaluated.  Wherever  it  was  possible  the 
moments  were  determined  separately  for  the  sections  of  positive  and  of 
negative  moment  shown  in  Fig.  12,  using  the  values  A,  f,  and  d,  shown  by 
observation  for  these  sections.  In  some  cases  it  was  necessary  to  use  an 
average  value  of  fj  for  both  sections  of  positive  moment  and  another  average 
value  for  both  sections  of  negative  moment.  In  some  cases  measured  values 
22 


ID 


06 


I  -  Total '  Moment (nu-- 


merical  sum  of  - 
pas  andneq.)  - 

2-  Tofal  Negative  _ 
Moment 

3-Neq.Moment;Col. 

-  Head  Section   - 
4-Totol  Posffire   - 

-  Moment 

5 -Po5.  Moment: 
Outer  Sect/on 

6 -Neq.  Moment; 
Mid  Section. 

7-  Pos.  Moment; 
Inner  Section. 


\ 


\ 


zo 

.18 

j 

1* 

j|./<? 

s/» 

J- 

r 
i 

"a/. 

> 

: 

7. 

n 

1-  Total  Moment(n 
merical  sumo) 
pos.  andneq.) 
Z-Tofal  Neoatin 
Moment 
3-NeqMoment,L 
head  Sect/or 
4  -Total  Positive 
Moment 
5  -Pos.  Moment, 
Outer5ection 
6-  Neg.  Moment 
Mid  Section 
7-  Pos.  Moment 
ImerSectioi 

\ 

^ 

\ 

V 

\ 

\ 

'b/ 

^ 

s 

^v 

\ 

> 

>v 

*? 

Sj 

*^v. 

'y* 

.04 
& 
n 

h 

f 

^ 

f>  j 

> 

\ 

( 

/ 

tf 

K 

K5 

i 

/ 

s 

i^ 

i 

c 

^ 

7 

*^ 

^^ 

TTtft 

-=s 

=^ 

•=3?f> 

ZOOM  3000040000  ZOOM  3000040000 

Computed  Average  1eny/e5tres5,  Ibpersqin. 

(«)  (6)  (a) 

FIG.    45.—  MOMENT    COEFFICIENTS    FOR    SANITARY    CAN 
UNCORRECTED  ;     (  6  )   CORRECTED. 

FIG.  46.  —  MOMENT  COEFFICIENTS  FOR  SHONK  BUILDING;     (a.)   UNCORRECTED 

(  6  )   CORRECTED. 


Computed  Aeraqe  Tensile  Stress  Ibpersq/n. 


BUILDING;      (a) 


of  d  were  not  available  and  it  was  necessary  to  use  what  appeared  to  be  the 
most  probable  values,  taking  into  account  the  total  thickness  of  the  slab, 
the  size  of  reinforcing  bars  and  the  number  of  layers  of  reinforcement.  It  is 
apparent  that  these  uncertainties  will. introduce  corresponding  uncertain- 
ties into  the  moment  coefficients,  but  the  errors  are  believed  to  be  no 
greater  than  the  errors  which  are  inevitably  involved  in  other  experimental 
work  of  an  equal  degree  of  complexity. 

Generally  the  computations  were  made  separately  for  the  moments  in 
the  directions  of  the  two  sides  of  the  panel,  and  were  combined  for  presenta- 
tion in  Fig.  43  to  48.  In  all  the  slabs  for  which  the  moment  coefficients 


MOMENTS  AND  STRESSES  IN  SLABS. 


91 


are  shown  the  panel  lengths  in  the  two  directions  were  so  nearly  the  same 
that  it  did  not  seem  desirable  to  show  separately  the  coefficients  for  the 
two  directions.  The  deficiencies  in  the  test  data  available,  and  the  unknown 
factors  which  affect  the  behavior  of  the  slab,  would  introduce  errors  which 
are  larger  than  the  difference  between  the  moments  in  the  two  directions. 
In  order  to  determine  experimentally  the  difference  in  moments  in  the  two 
directions,  when  the  spans  are  so  nearly  equal,  a  large  number  of  tests 
would  be  required,  and  the  deficiencies  in  the  test  data  would  have  to  be 


/-  Total  Moment(nu- 


pos.andnea.) 
<?-  Total  Negative 
Moment. 

3-  Neq.Moment;Col. 
Head  Section. 

4-  Tofa/ Positive 
Moment 


•5  -Fbs.  Moment; 

OuferSection. 
6-Neq.Moment- 

Mid  Section 
7-  fas.  Moment: 

Inner  Section 


FIG.    47.- 


ZOOM  30000  40000  30000 

Computed  Average  Tensi/eSfress  /bpersq/n 
(a)  (0) 

-MOMENT    COEFFICIENTS    FOR    BELL    STREET    WAREHOUSE; 
UNCORRECTED;    (6)  CORRECTED. 


(a) 


supplied.  One  test,  that  of  the  Larkin  Building,  was  available,  in  which 
the  differences  in  the  size  of  the  panels  was  considerable;  but  in  this  test, 
except  in  the  lower  loads,  the  number  of  panels  loaded  was  not  sufficient  to 
give  corresponding  conditions  in  the  two  directions,  and  the  coefficients  for 
that  slab  are  not  presented. 

The  computed  average  tensile  stress,  shown  as  abscissas  in  Fig.  43  to 
48,  were  determined  from  the  equation 


\ 


Ajd 


in  which 


W  is  the  total  panel  load,  live  and  dead,  and    2  Ajd    is  thp  sum  of  the 


92 


MOMENTS  AND  STRESSES  IN  SLABS. 


values  of  Ajd  for  the  sections  shown  in  Fig.  12.  The  values  of  Ajd  for 
these  sections  are  given  in  Table  XIII. 

It  was  shown  in  Art.  16  that  the  relation  between  the  observed  stress 
and  the  computed  stress  in  the  reinforcement  is  affected  by  variations  in  the 
value  of  n  (that  is,  of  the  modulus  of  elasticity  of  the  concrete,  since  that 
of  the  steel  is  practically  constant).  Equations  (5)  and  (6),  Art.  16,  show 
this  effect  below  and  above  the  load  at  which  the  concrete  cracked.  Corre- 
spondingly the  value  of  the  corrected  moment  coefficient  will  be  affected  by 
these  variations  in  n. 

Fig.  49  has  been  prepared  to  show  the  effect  of  a  variation  in  the  value 
of  n  on  the  coefficients.  Small  circles  indicate  the  coefficients  for  the  value 
of  n,  which  was  used  in  obtaining  the  corrected  moment  coefficients  shown 
in  Fig.  43  to  48.  This  is  the  average  value  of  n  for  the  stone  and  the 


JJ|M-/2" 
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Computed  Oreraqe  Tensi/e  Jtnsss,  /b.perjq.in. 

(a)  (6) 

FIG.  48. — MOMENT  COEFFICIENTS  FOR  WESTERN  NEWSPAPER  UNION  BUILDING  ; 
(a)  UN  CORRECTED;    (6)  CORRECTED. 

gravel  concretes  of  the  beams  used  in  the  tests  by  the  U.  S.  Geological 
Survey.  (See  Table  VIII,  Part  III).  The  smooth  curves  show  the  cor- 
rected coefficients  which  have  been  found  by  using  a  range  of  values  of  A  in 
the  determination  of  the  corrections. 

It  will  be  seen  that  in  order  to  bring  the  coefficients  to  the  theoretical 
value  of  %  the  value  of  n  for  the  Purdue  tests  would  be  about  9  and  that 
for  the  Sanitary  Can  Building  and  the  Shonk  Building  would  be  about  11.5. 
While  it  cannot  be  stated  that  these  are  the  correct  values  of  n  for  the 
concrete  in  these  slabs,  it  seems  reasonable  to  believe  that  they  are  more 
nearly  correct  than  the  value  of  7,  which  was  used  in  computing  all  cor- 
rected moment  coefficients.  The  available  data  from  the  slab  tests  strengthen 
this  belief,  but  the  reliability  of  the  test  data  which  bear  on  this  subject 
was  not  sufficient  to  justify  introducing  into  the  computation  of  the  cor- 
rected moment  coefficients  the  experimental  values  of  n,  determined  in  the 
tests  of  these  structures. 


MOMENTS  AND  STRESSES  IN  SLABS. 


93 


It  will  be  seen  in  Figs.  43  to  48  that  the  uncorrected  moment  coefficients 
are  all  less  than  the  theoretical  value,  %.  For  the  lower  computed  stresses 
the  corrected  coefficients  are  generally  in  excess  of  ys,  but  for  the  highest 
computed  stress,  that  is,  for  the  highest  load  applied,  the  average  coefficient, 
0.111,  is  less  than  the  theoretical  coefficient. 

The  fact  that  for  the  lower  computed  stresses  the  corrected  moment 
coefficients  generally  were  higher  than  the  theoretical  value,  indicates  that 
too  large  a  correction  factor  was  used.  No  means  of  knowing  how  much 
the  factor  used  was  in  error  is  evident,  but  the  fact  that  as  the  computed 
stress  increases  the  uncorrected  and  the  corrected  moment  coefficients 
approach  each  other  in  value  seems  to  reduce  the  uncertainties  as  to  the 
correct  values  of  the  moment  coefficients  to  a  narrower  margin  than  that 
which  has  limited  the  usefulness  of  practically  all  the  field  tests  that  have 
ever  been  made  on  reinforced-concrete  slabs.  On  the  average,  the  agreement 


0        Z       4        68        IO       IZ      14 

Values  of  n 

FIG.  49. — EFFECT  OF  VARIATION  IN  MODULUS  OF  ELASTICITY  ON  VALUE  OF 
MOMENT  COEFFICIENTS  AT  Low  LOADS. 


of  the  corrected  coefficients  for  the  higher  computed  stresses  with  the  result 
found  by  analysis  in  Part  II  is  sufficiently  close  to  warrant  the  belief  that, 
if  all  the  sources  of  error  in  the  measurement  of  deformations  and  in  the 
interpretation  of  test  results  could  be  removed,  the  analysis  and  the  tests 
would  be  in  substantial  agreement.  It  is  pointed  out  in  Art.  20,  in  discuss- 
ing the  results  of  the  test  of  a  slab  in  which  there  were  beams  on  the 
boundary  lines  of  the  panels,  that  although,  for  the  lower  loads,  the  test 
results  were  in  fair  agreement  with  the  analysis  of  that  type  of  slab,  there 
was,  for  the  higher  loads,  an  accommodation  of  the  slab  to  the  conditions 
imposed  upon  it,  which  made  the  slab  capable  of  carrying  a  much  greater 
ultimate  load  than  is  accounted  for  by  equating  the  applied  bending  moment 
and  the  apparent  resisting  moment.  On  account  of  the  lack  of  tests  of  flat 
slabs  carried  to  the  point  of  failure,  and  in  which  the  design  was  such  as 
to  preclude  failure  from  some  other  cause  than  bending,  it  cannot  be  stated 
that,  to  the  same  extent,  a  similar  source  of  additional  strength  is  present 
in  flat  slabs  as  was  present  in  the  Waynesburg  slab,  which  was  supported  on 


94  MOMENTS  AND  STRESSES  IN  SLABS. 

four  sides.  There  are  some  indications,  however,  that  there  was  greater 
strength  in  the  flat  slabs  than  appears  from  the  conclusion  that  the  test 
results  and  the  analysis  of  moments  are  in  fair  agreement.  Some  indica- 
tion of  this  greater  strength  is  seen  in  the  description  of  the  Purdue  tests, 
Appendix  B. 

24.  FACTOB  OF  SAFETY.  Table  XIII  gives  summarized  data  of  all  the 
tests  studied.  The  purpose  of  this  table  is  to  give  the  best  estimate  of  what 
the  factor  of  safety  against  failure  of  the  structure  would  have  been  if 
sufficient  load  had  been  applied  in  each  case  to  produce  failure.  For  all 
the  tests  reported,  except  the  Purdue  tests,  the  factor  of  safety  is  an  esti- 
mated quantity  which  gives  the  ratio  of  the  estimated  maximum  load  which 
the  slab  would  carry  to  the  design  load  (sum  of  dead  and  live  load),  com- 
puted as  indicated  in  the  table.  Although  the  factor  of  safety  is  shown  for 
three  moment  coefficients,  the  design  load  is  shown  for  only  one  coefficient. 
The  design  load  will  be  inversely  proportional  to  the  coefficient  used  in  the 
design.  For  the  Purdue  test  slabs  the  factors  of  safety  given  are  the  ratios 
of  the  load  actually  applied  to  the  slab  to  the  design  load  shown  in  the 
table.  The  "average  observed  stress,  /„"  of  the  table,  therefore,  has  no  sig- 
nificance for  those  slabs,  but  they  are  given  because  of  their  value  as  mat- 
ters of  general  information  concerning  the  tests.  The  stresses  for  the 
maximum  load  of  872  Ib.  per  sq.  ft.,  on  slab  J,  were  not  reported,  and  the 
average  observed  stress  given  for  that  slab  is,  therefore,  that  for  the  load 
of  664  Ib.  per  sq.  ft.  (live  and  dead),  the  highest  load  for  which  the  meas- 
ured stresses  were  reported.  For  all  other  tests  the  average  observed 
stress  given  in  the  table  is  that  for  the  "maximum  test  load"  given  in  the 
table.  The  average  observed  stresses,  as  reported,  were  roughly  weighted 
to  take  account  of  the  distribution  of  the  gage  lines  over  the  sections  of 
maximum  stress  due  to  negative  moment  and  positive  moment.  For  the 
Purdue  tests  this  weighting  was  not  necessary,  since  the  stresses  were  meas- 
ured in  all  the  bars  crossing  these  sections. 

The  "estimated  dead  load  stress"  of  Table  XIII  is  given  by  equation 
(1)  or  equation  (3)  of  Part  III,  in  which  ys  is  the  value  of 

,  I 

Js 


2  Ajd 

In  this  equation  W  is  taken  as  the  total  dead  load  of  the  panel.  Since  the 
value  of  /  was  generally  below  the  points  in  the  curves  of  Fig.  31,  which 
represent  the  cracking  of  the  concrete,  equation  (  1  )  was  generally  used  in 
estimating  the  dead  load  stress  rather  than  equation  (  2  )  . 

In  estimating  the  maximum  load  for  slabs  it  was  assumed  that  the 
yield  point  of  the  steel  was  40,000  Ib.  per  sq.  in.,  and  that  failure  would 
have  occurred  at  a  computed  average  stress,  which  is  the  same  as  the  com- 
puted stress  given  in  Fig.  31  at  the  intersections  of  the  straight  lines,  of 
equation  (  3  )  ,  part  III,  with  the  locus  representing  the  equation  /  =  40,000 
(0.82-|-7p)  given  in  Fig.  31.  The  assumption  of  40,000  Ib.  per  sq.  in.  as 


MOMENTS  AND  STRESSES  IN  SLABS. 


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96  MOMENTS  AND  STRESSES  IN  SLABS. 

the  yield  point  was  made  in  order  to  bring  all  the  tests  to  a  common  basis 
for  the  purpose  of  comparison  and  in  order  to  obtain  a  factor  of  safety  not 
higher  than  might  be  expected  if  reinforcement  having  that  yield  point 
were  used.  Obviously  reinforcement  having  a  yield  point  of  55,000  Ib.  per 
sq.  in.  should  be  expected  to  give  a  higher  factor  of  safety  than  that  having 
a  yield  point  of  40,000  Ib.  per  sq.  in.,  when  the  working  stress  in  tension 
is  16,000  Ib.  per  sq.  in.  in  both  cases. 

The  considerations  in  the  two  preceding  paragraphs  indicate  that  the 
true  factor  of  safety  was  probably  higher  than  the  values  given  in  Table 
XIII.  On  the  other  hand  it  must  be  recognized  that  the  design  load  found 
with  the  use  of  the  measured  depth  d  would  be  smaller  and  the  factor  of 
safety  somewhat  larger  if  the  depth  d  assumed  in  the  design  of  the  slabs 
were  used  for  the  computations  of  factor  of  safety.  This  is  due  to  the  fact 
that,  on  account  of  misplacement  of  the  reinforcement,  the  measured  depth 
to  the  reinforcement  is  usually  somewhat  less  than  the  depth  used  in  design. 

The  average  factor  of  safety  for  the  structures  reported  in  Table  XIII, 

estimated  on  the  basis  of  a  moment  coefficient  of  0.1067  Wl   1 1  -  -  -l  2* 

\        3  l) 

is  3.23  and  that  for  the  moment  coefficient  of  0.09  Wl      (\  _  ?  2\  2*  is  2.72. 

\        3  l) 

It  is  to  be  noted  that  for  the  tests  which  were  carried  to  destruction  of  the 
slab,  or  nearly  so  (the  two  Purdue  tests  and  Western  Newspaper  Union 
Building  test)  the  values  were  above  these  average  values. 

It  has  been  established  by  tests  that  the  maximum  load  on  a  simple 
beam  occurs  at  a  tensile  stress  only  slightly  greater,  say  10  per  cent,  for 
steel  of  structural  grade,  than  the  yield  point  of  the  steel  in  tension  and  not 
at  the  ultimate  strength  of  the  steel.  This  is  true  not  only  for  reinforced 
concrete  beamsf  but  also  for  steel  beams. $ 

Kecognizing  this  fact,  and  at  the  same  time  using  a  working  stress  of 
16,000  Ib.  per  sq.  in:  in  steel  of  structural  grade,  whose  yield  point  is  33,000 
Ib.  per  sq.  in.,§  is,  in  fact,  recognizing  the  sufficiency  of  a  factor  of  safety 

of  about  2.25.    Based  upon  the  use  of  a  moment  of  0.09  Wl    M  _  ^  £. \  2   for 

\  O      i  / 

design,  the  slabs  reported  in  Table  XIII  would,  on  the  average,  develop  this 
factor  of  safety  of  2.25  even  though  they  were  to  have  failed  at  loads 
approximately  15  per  cent  greater  than  the  loads  which  were  applied.  To 
one  familiar  with  the  behavior  of  the  structures  listed  in  Table  XIII,  or 
with  similar  structures  during  and  after  the  tests,  it  is  obvious  that  they 
would  have  carried  at  least  that  much  additional  load. 


*  These  are  the  total  moments  recommended  by,  respectively,  the  Joint  Committee 
on  Concrete  and  Reinforced  Concrete  and  the  American  Concrete  Institute  Committee 
on  Reinforced  Concrete  and  Building  Laws. 

t  Art.  18  and  Fig.  36;  also  A.  N.  Talbot  Bull.  1.  Univ.  of  111.  Eng.  Exp.  Sta. 
(1904),  p.  27;  Turncaure  and  Maurer  "Principles  of  Reinforced  Concrete  Construc- 
tion," 2nd  ed.  (1909),  p.  142. 

t  H.  F.  Moore  Bull.  68,  Univ.  of  111.  Eng.  Exp.  Sta.  (1913),  p.  14. 

$  Standard  Specifications  for  billet  steel  concrete  reinforcement  bars  A  15-14.  A.  S 
T.  M.  Standards.  1913.  D.  148. 


MOMENTS  AND  STRESSES  IN  SLABS.  97 

It  seems  certain,  therefore,  that  for  the  flat  slabs  under  discussion 
the  factor  of  safety  against  failure  in  bending,  based  on  a  total  moment  ol 

0.09  Wl(l  -  -  y)2,  is  at  least  as  great  as  that  which  can  be  counted  upon 

in  the  most  elementary  flexural  unit,  the  simple  beam  built  of,  or  reinforced 
with,    steel    of    structural    grade    and    designed    with   the   usual   working 

stresses.    Based  upon  a  total  moment  of  0.125  Will  —  _     )  *  the  factor  of 

\  u    1 1 

safety  is  correspondingly  greater. 

An  effort  was  made  to  obtain,  from  failures  of  flat  slabs  in  service, 
information  as  to  what  were  the  serious  weaknesses  in  the  design.  A  study 
of  a  letter  from  Edward  Godfrey*  citing  29  failures  of  reinforced-concrete 
buildings  brings  out  the  fact  that  all  of  the  building  floors  there  discussed, 
including  flat  slabs  among  others,  failed  during  construction  or  as  a  result 
of  a  severe  fire.  While  some  of  the  failures  during  construction  may  have 
been  partly  due  to  deficiencies  in  the  design  the  lack  of  proper  safeguards 
in  construction  were  so  evident  and  so  important  that  with  the  meager 
data  available  any  effort  to  analyze  the  failures  would  be  long  drawn  out 
and  largely  speculative.  The  fact  that  the  majority  of  the  failures  referred 
to  occurred  during  winter  weather  or  in  structures  which  had  no  oppor- 
tunity to  cure  properly  in  warm  weather  is  in  itself  sufficient  indication 
that  poor  construction  conditions  contributed  largely  to  the  failure.  To 
attempt  to  guard  against  abusive  lack  of  safeguards  in  construction  by 
severe  requirements  for  design  would  be  ineffective  and  prohibitively 
extravagant  and  would  itself  encourage  the  omission  of  these  safeguards. 
There  probably  are  cases  in  which  deficiencies  of  design  have  caused 
trouble  in  flat  slab  structures,  but  cases  of  this  kind,  with  data  of  the 
design  and  loading  sufficient  to  be  of  value  in  the  study  of  the  factor  of 
safety,  have  not  been  found  in  the  preparation  of  this  paper. 

25.  SHEARING  STRESSES.  In  Table  XIII  the  maximum  shearing  stresses 
on  a  vertical  section  at  a  distance  d,  from  the  edge  of  the  column  capital 
for  the  tests  summarized  in  that  table  are  given.  Those  shearing  stresses 
were  computed  on  the  basis  of  a  depth  jd.  The  highest  shearing  stress 
developed  was  in  the  case  of  the  Western  Newspaper  Union  Building.  The 
reportf  does  not  indicate  that  the  test  developed  any  weakness  in  shear. 
In  order  to  develop  a  factor  of  safety  as  high  in  shear  as  the  estimated 

factor  of  safety  in  bending  based  upon  the  total  moment,  0.1067  Will  — 
~  —  j J,  the  shearing  stress  in  the  section  under  question  could  have  been  at 

O      i  j 

least  87  Ib.  per  sq.  in.  at  the  design  load.     Using  the  factor  of  safety  in 
bending  for  the  total  moment  of  0.09  Win  _  ~  £. ) »  the  shearing  stress  for 

the  design  load  could  have  been  at  least   102  Ib.  per  sq.  in.  There  were 
some  indications  that  the  failure  of  slab  J  may  have  been  due  to  shear. 


*  Edward  Godfrey,  "An  open  letter  to  W.  A.  Slater,"  Concrete,  February,  1921. 
t  Univ.  of  111.  Eng.  Exp.  Sta.  Bulletin  106. 


98  MOMENTS  AND  STRESSES  IN  SLABS. 

The  allowable  shearing  stresses  at  the  design  load  which  would  give  the 
same  factors  of  safety  as  those  shown  for  the  moments  would  have  been 
49  Ib.  per  sq.  in.  and  58  Ib.  per  sq.  in.  respectively. 

The  shearing  stresses  are  not  given  for  the  Jersey  City  Dairy  building 
nor  for  the  International  Hall.  This  is  because  the  loads  reported  in  this 
table  were  not  such  as  to  give  uniform  shear  around  the  perimeter  of  the 
column  capital,  and  it  is  not  known  what  the  maximum  shear  was. 

VI.— SUMMARY. 

The  theoretical  analysis  deals  with  slabs  of  homogeneous  perfectly 
elastic  material  and  of  uniform  thickness.  Two  types  are  considered  in 
particular:  slabs  supported  on  four  sides  and  flat  slabs  supported  on 
columns  with  round  capitals.  Moment  coefficients  derived  by  principles  of 
equilibrium  and  continuity  are  shown  in  the  diagrams  and  tables. 

The  close  agreement  between  the  moment  coefficients,  determined  by 
several  investigators,  in  slabs  supported  on  four  sides,  is  an  indication  that 
dependable  methods  are  now  available  by  which  homogeneous  elastic  slabs 
may  be  analyzed. 

In  the  analysis  of  flat  slabs  the  moment  sections  used  in  the  Joint 
Committee  report  of  1916  were  found  suitable  for  the  purpose  of  stating  the 
resultant  moments.  In  a  square  interior  panel  of  a  uniformly  loaded 
floor  slab,  with  a  large  number  of  panels  in  all  directions,  the  percentages 
of  the  total  moment  (or  sum  of  positive  and  negative  moments)  which 
are  resisted  in  the  column-head  sections,  mid-section,  outer  sections,  and 
inner  section  are  found  to  be  nearly  independent  of  the  size  of  the  column 
capital. 

A  study  is  made  of  unbalanced  loads,  for  example,  loads  in  rows; 
unbalanced  loads  are  found  to  produce  large  moments  in  the  slabs  if  the 
columns  are  slender,  and  large  moments  in  the  columns  if  the  columns  are 
rigid.  Moment  coefficients  are  stated  also  for  various  cases  of  oblong 
panels,  wall  panels,  and  corner  panels. 

The  tests  of  slabs  supported  on  four  sides  indicate  that  when  the 
deformations  increase,  certain  redistributions  of  moments  and  stresses  take 
place,  with  the  result,  in  general,  that  the  larger  coefficients  of  moments 
are  reduced.  The  ultimate  load  is  found  to  be,  in  general,  larger,  and  in 
some  cases  much  larger,  than  would  be  estimated  on  the  basis  of  the  theo- 
retical moment  coefficients  and  the  known  strength  of  beams  with  the  same 
ratio  of  steel. 

When  the  moments  of  resistance  of  the  observed  stresses  in  the  rein- 
forcement in  flat  slabs  were  multiplied  by  the  ratio  of  the  applied  moment 
in  simple  beams  to  the  resisting  moment  of  the  observed  stress,  corrected 
moments  for  the  slabs  were  obtained  which,  in  comparison  with  the  results 
of  the  analysis  of  flat  slabs  presented  in  Part  II,  were  (a)  much  greater 
for  the  lower  loads  than  for  the  higher  loads,  (b)  greater  for  the  lower 
loads  than  the  theoretical  moments  and  (c)  slightly  less  for  the  higher 
loads  than  the  theoretical  moments. 


MOMENTS  AND  STRESSES  IN  SLABS.  99 

If  the  effect  of  the  difference  in  the  modulus  of  elasticity  for  the  slabs 
from  that  for  the  beams  on  which  the  comparison  is  based  could  be  elimi- 
nated it  seems  that  the  agreement  between  the  analysis  and  the  tests  would 
be  fair.  Such  information  as  is  available  on  the  effect  of  the  modulus  of 
elasticity  on  the  results  points  in  the  direction  stated. 

The  average  value  of  the  estimated  factor  of  safety  for  the  slabs 
studied  was  3.23  for  the  working  loads  based  upon  the  moment  coefficients 
recommended  by  the  Joint  Committee  on  Concrete  and  Reinforced  Concrete 
and  2.72  for  the  working  loads  based  on  the  coefficients  recommended  by  the 
American  Concrete  Institute. 


APPENDIX   A. 

DETAILS   OF   THE    ANALYSIS    OF    HOMOGENEOUS    PLATES. 
BY  H.  M.  WESTERGA.ARD. 

Al.  Notes  referring  to  Art.  7.  Solutions  of  the  Differential  Equation  of 
Flexure  for  Slabs  Supported  on  Four  Sides,  (a)  Rectangular  slabs 
with  simply  supported  edges.* 

The  analysis  of  the  rectangular  slab  with  simply  supported  edges  is 
simplified  by  assuming  certain  special  values  of  the  dimensions,  of  the 
clastic  constants,  and  of  the  load:  namely, 

a  =  "f  =  long  span,  in  the  direction  of  x; 

b  =   oc    TT  =  '_   =  short  span,  in  the  direction  of  y; 

M 
w  =  —  .    =  load  per  unit-area; 

7T2 

£7  =  1;   Poisson's  ratio  K  =  0. 

The  origin  of  the  coordinates  x,  y  is  at  the  center  of  the  slab. 

With  these  special  values,  one  finds  wb-  =  1  ;    that  is,  the  moments 

M 
become  equal  to  the  coefficients  of  moment,  -ii±_  .      Since  El  =  1  and  K  =  0, 

wb2 

the  moments  or  coefficients  of  moment  become  numerically  equal  to  the 
curvatures.  The  expediency  of  analyzing  with  a  Poisson's  ratio  equal  to 
zero  was  discussed  in  Articles  6  and  8,  where  also  methods  of  modifying 
the  results  when  Poisson's  ratio  has  some  other  value  were  indicated. 
(See  equations  (15)  to  (18)  in  Art.  6  and  Fig.  10(b)). 

In  order  to  solve  Lagrange's  equation  of  flexure  (  (11),  (12),  or  (19) 
in  Art.  6), 


where  $4          5"        g< 

AA=r-4  +  2  rj^-i  +  7-4 
dx*        t>x  &y        by 


*  Navier's  solution  is  used  here  (see,  for  example,  A.  E.  H.  Love,  Mathematical 
Theory  of  Elasticity,  ed.  1906,  p.  468).  Levy's  solution  (Love,  p.  469)  was  used  by 
Nadai  in  dealing  with  the  same  problem  (see  the  historical  summary  in  Art.  4,  foot- 
note 36). 


100  MOMENTS  AND  STRESSES  IN  SLABS. 

8* 
the  term    —    is  expressed  by  a  double-infinite  Fourier  series,  as  follows: 

mtn 

8      '6/3    m    £     ~(~U 

(m.n-  1,3,5,7 )  (47) 

This  expression  applies  at  all  points  of  the  slab  except  at  the  edges.  If 
the  load  w  had  consisted  of  only  one  of  the  terms  in  (47),  the  solution  of 
Lagrange's  equation  would  be:  z  equal  to  a  similar  term,  which  is  equal  to 
a  constant  times  the  load  at  the  particular  point.  With  w  equal  to  the 
complete  series  ( 47 ) ,  the  solution  of  ( 46 )  becomes : 


CO5  (48) 

This  solution  satisfies  equation   (46),  as  may  be  verified  by  substitution, 
and  it  satisfies  also  the  boundary  conditions,  that  at  the  edges     z     -     0, 

5>2  $2 

—  -  0,  and  —  Z  -  0.  The  deflections,  therefore,  are  expressed  correctly  by 
Sz2  8j/2 

equation  (48). 

By  double  differentiations  of    (48),  one  finds  the  bending  moments 
(according  to  (20)  in  Art.  6)  to  be: 

M        S'z     I6tf  *    n   -{-tfFm 

'-  Sx'  =  -fig.  g  .  nlm'+fWy  C°5  ™  005Pny  (49) 

and 


(50) 
and  the  torsional  moment  (according  to  (21)   in  Art.  6)   to  be 

rl  .,Q3  m  ,    .  \al^D 

Mrx-T-r  =  —  j  2123  -.  —  ,  ai   ,,z  sin  mx  sinBny 

6x«5y    TT    i.3»t,3"(m+J)n)  (51) 

The  moment   coefficient   at   the   center   of   a  square   slab   is   found   by 
substituting  x  =  y  =  0  and  ft  =  1  in   (49)   or   (50),  and  it  is 

M_J6_  «£    -H)mlam 
~ 


16 


5+\f    3(9+9)*    5( 

_J  _  N  1 

7(1+49)*)         "J 


.^•0.2245-00369,  (52) 

as  shown  on  the  diagram  in  Fig.  3 (a).  This  calculation  is  typical;  other 
coefficients  shown  in  Fig.  3 (a)  were  computed  in  the  same  way.  It  may  be 
noted  that  in  (52)  the  terms  of  the  double-infinite  series  are  arranged  in 


MOMENTS  AND  STRESSES  IN  SLABS.  101 

groups  with  m  -(-  «  =  2,  4,  6,  8,  10  .........  ,  respectively,  and  thus  the 

double-infinite  series  is  transformed  into  a  single-infinite  series. 

The  moment  M  a  across  the  diagonal  at  the  corner  in  a  square  slab 
is  numerically  equal  to  the  torsional  moment  M  at  the  point,  defined  by 
(51)  ;  that  is, 


(53) 

The  moment  M  at  the  corner  of  a  rectangular  slab  across  a  line 
making  angles  of  45  degrees  with  the  sides,  is  determined  by  an  expression 
similar  to  (53),  but  containing  the  ratio  fi  of  the  long  span  to  short  span. 
In  the  limiting  case  in  which  ft  =  oo  (or,  ex  =0),  this  expression  may 
be  reduced  to  the  form, 


(54) 
which  is  the  value  shown  at  the  left-hand  edge  in  Fig.  3  (a). 

(b)    Infinitely   long   strip,   extending   from   x  —  0   to  x  =  oo  between 

the  simply  supported  edges  y  —  ±  -  ;      fixed  edge  along  the  i/-axis.     This 

£t 

slab  is  a  special  rectangular  slab  with  the  spans    a   **    oo,    b   •»    jf.      Load 

w--;     v*»  =  -  .   K  =  0;  El  =  1. 
4  4 

The  solution  of  Lagrange's  equation, 


is  written  in  the  form 

z  =  z±  -\-  2, 

where  c,  is  the  deflection  at  the  point  (x,y)  when  the  support  at  the  t/-axis 
is  removed,  so  as  to  make  the  edge  deflect  freely.  The  remainder  zt  is  the 
amount  which  is  added  when  external  forces  applied  at  the  free  edge  at  the 
y-axis  make  the  deflections  and  the  slopes  at  this  line  again  equal  to  zero. 
2,  is  the  deflection  of  a  simple  beam  with  a  span  equal  to  TT  ,  and  may  be 
expressed  as  a  polynomial  in  y,  but  may  also  be  expressed  by  the  Fourier 
series 

21  =  cos  y  —  —&  cos3?/  -f  —  B  cosSy  -  ^  cos7y  +  ......  , 

as  may  be  verified  by  comparison  with  the  expression  for  the  load 

w  =  -  =  cos  y  --  cos3?/  +  -  cos5y  —  -  cos7y  +  ....... 

4  357 

The  deflection  z2  must  satisfy  the  following  conditions:    at  all  points, 

Sz 
A  A  zz  =  0;  at  the  short  edge,  z2  =  zi  and  —  =  0;  at  the  long  edges,  z2  -  0 

82z 
and  —  =0.     These  conditions  are  satisfied  by 


,  —  (-\fy- 

„    (/-t-mxje        N      s — co5  my 


102 


MOMENTS  AND  STRESSES  IN  SLABS. 


i 

Fince   —  =  0,  the  bending  moment  along  the  z-a\is  becomes 


(55) 


When  x  =  O,  this  moment  becomes  M a«  -      2     ~^T 


m3 


=     —    —  Ri'r 


IT'  .          '  •"*  1 

wb*  -  — ,  the  corresponding  momsnt  coafficient  becomes   ~?  =  —  _ ,         as 
4  u-'O2  8 

shown  at  left-hand  edge  in  Fig.  6  (a).  The  series  (55)  converges  rapidly. 
Values  of  Max  were  computed  for  x  =  05,  1.0,  2.0,  3.0,  and  4.0;  the  greatest 
positive  value,  0.1339,  which  was  found  with  x  =  2.0,  gives  the  coefficient 

M 

— —  =  0.0173,  as  shown  at  the  left-hand  edge  in  Fig.  5  (a). 
u-b2 

y  y 

5 imply  supported 


Fiu.  Al. — RECTANGULAR  SLAB. 


(c)     The  rectingulnr  slab  shown  in  Fig.  A  I.     Two  parallel  edges  are  fixed 
and  tv.'o  parallel  edges  simply  supported       Simple  span  =•  "";  fi*ed  span  =  I. 


Load  w  =  —  ;  when  /  >TT,  then  ivb2  =  —  ;     when   I 
44 


,   then   wb2   =  -  I2; 
4 


£7=1;  tf  =  0. 

The  solution  given  here  is  essentially  Levy's  solution.*  The  procedure 
is  essentially  the  same  as  in  the  preceding  case,  (b),  which  is  the  special 
case  in  which  I  =  oo  . 


Lagrange's  equation 


AAz  = 


(56) 


is  solved  by 

z  =  zi  +  z-i  (57) 

where,  as  in  case   (b),  zl  is  the  simple-beam  deflection,  obtained  when  the 


*  See    the   historical    summary,    Art.    4,    footnote    13,    or   A.    E.    H.    Love,    Mathe- 
natical  Theory  of  Elasticity,  Ed.  1906,  p.  469. 


MOMENTS  AND  STRESSES  IN  SLABS.  103 

edges  y  and  y'  are  removed,    c,  may  be  expressed  either  by  a  polynomial  in 
y  or  by  the  Fourier  series 

Zi  =  cosij  -  -  cos'3y  +  —  cos5y  -  ......  (58) 

The-  remaining  part  cs  of  the  deflection  must  satisfy  the  following  condi- 
tions: 

AA2,  =  ()  at  all  points;  (59) 

8«, 

Z2  =  0,  s—  =0     at  the  simply  supported  edges;  (60) 

£  5 

s—  =  0,  £~Y  =  0  at  the  fixed  edges;  (61) 

Zi  =  —  z\  at  the  fixed  edges.  (62) 

The  conditions   (59)    and   (60)   may  be  satisfied,  as  may  be  verified  by 
differentiation,  by  an  expression  of  the  form 

S     (-D^r 
z^Z  -—r-$mcoSmy  (63) 

where 


e+xe  (64) 

in  which    Km  and  k      are  cimstants.     This  solution  will  satisfy  the  condi- 
tion  (61)   when 


>+(ml-l)e-m  (65) 

and  it  will  satisfy  the  condition  (62)  when 

K  =  _  L_ 

/  +  (»+rnh-TOl)e  (66) 

The  deflections,  then,  are  defined  completely  by  equations   (57)    (58),  and 
(63)   to   (66). 

The  bending   moments  in  the  x-  and  y-  directions  may  be  found  by 
double  differentiations  of   (57),   (58),  and   (63).     One  finds 

S'z       Szz,    pf-i)2^  SX 
M*  =  -  TT  -  -  r~t  =  ^  —  j—  -r~r  cos  my 
^          J      ..       3 


+  e         cos  my 


and 


*  v      J5v*  =  ~Jt   z  ~  JT*  =  T  \~A  ~y  I         — r   '  '       ( 68) 

The  series  (67)   and   (68)   converge  rapidly.     When  these  series  are  used  in 
connection   with   formulas    (64)    to    (66)    they  are   suitable   for  numerical 


104  MOMENTS  AND  STRESSES  IN  SLABS. 

computations,  and  they  were  used  in  determining  the  points  shown  by 
small  circles  in  Fig.  4 (a),  Fig.  5 (a),  and  Fig.  6 (a). 

With  i/=0  and  1=  oo,  formula  (67)  becomes  the  same  as  (55). 

(d)   Slabs  with  four  ficced  edges. 

The  approximate  moment  coefficients  for  square  slabs,  represented  in 
Fig.  7  and  Fig.  8  by  points  marked  with  circles,  were  determined  by  approx- 
imate expressions  which  contain  trigonometric  and  exponential  functions 
of  x  and  y;  they  are  somewhat  similar  in  form  to  those  applied  in  the 
preceding  cases.  Neither  Navier's  nor  Levy's  solution  applies  directly  to 
the  slab  with  four  fixed  edges.  Ritz's  method,  which  was  used,  for  example, 
by  Nadai  and  Paschoud  in  analyses  of  fixed  slabs,  is  found  to  lead  to  suit- 
able solutions  of  the  problem.* 

A2.  Theory  of  Ring  Loads,  Concentrated  Couples,  and  Ring  Couples. 
Certain  concentrated  loads,  each  consisting  of  a  group  of  forces  within  a 
small  area,  were  introduced  in  Articles  8  and  9,  where  procedures  of 
analyses  of  flat  slabs  were  outlined,  and  where  the  results  of  these  analyses 
were  presented.  The  loads  introduced  are:  the  ring  loads,  which  were 
defined  in  Art.  8,  and  which  are  used  in  the  analysis  of  the  normal  square 
interior  panel  of  a  uniformly  loaded  flat  slab;  and  the  concentrated 
couples  and  ring  couples,  which  were  defined  in  Art.  9,  and  which  are  used 
in  the  study  of  unbalanced  loads. 

By  the  use  of  the  concentrated  loads  in  the  analysis  of  flat  slabs  a 
procedure  is  followed  which  has  general  applicability,  and  which  was  used 
in  one  form  in  the  preceding  article  (in  cases  (b)  and  (c)  )  : 

Lagrange's  equation, 

AA*  =  Lz__*2  „-,  (12) 

is  solved,  for  the  given  loads  and  boundary  conditions,  by  expressing  the 
deflection  in  the  form 

z  =  *0  +  2*m  (69) 

where  z0  satisfies  (12),  without  necessarily  satisfying  the  boundary  condi- 
tions, while  each  function  z  satisfies  the  equation 

AA2m  =  0,  (70) 

which  is  Lagrange's  equation  for  w  =0. 

The  deflection  z  in  (69)  may  be,  for  example,  the  deflection  of  the 
point-supported  slab  under  the  load  w.  Then,  z  may  be  the  deflection  due 
to  one  of  the  concentrated  loads,  acting  alone  on  the  slab  at  a  point  of 
support,  with  the  surrounding  supports  removed.  By  introducing  one  such 
concentrated  load  at  each  point  of  support  or  panel  corner,  and  by  adding 
the  deflections  due  to  all  of  the  loads,  one  forms  the  series  (69) ,  which  may 
be  an  infinite  series,  by  differentiation  of  which  one  may  obtain  corre- 
sponding series  for  the  moments.  The  concentrated  loads  must  be  so 
selected  that  all  of  the  loads,  including  the  applied  load  w,  will  cause  the 


•  Sec  the  historical  summary,  Art  4,  footnotes  26.  34,  and  36. 


MOMENTS  AND  STRESSES  IN  SLABS.  105 

point-supported  slab  to  deflect,  outside  the  circles  marked  by  the  edges  of 
the  column  capitals,  exactly  as  the  slab  deflects  which  is  supported  on 
column  capitals  and  loaded  by  w. 

In  the  theory  of  the  concentrated  loads,  now  to  be  presented,  it  is 
assumed  at  first  that  only  one  concentrated  load  acts  on  the  slab.  Then, 
groups  of  loads  are  considered.  The  slab  is  assumed  to  extend  indefinitely 
in  all  directions.  Furthermore,  let: 

K  =  Poisson's  ratio  =  0,  as  before; 

r  =  yx2  _j_  yi  •=•  radius  vector  measured  from  the  origin  of  the  co- 
ordinates. 

(a)   Ring  loads. 

Lagrange's  equation 

AAz  =  0,  (71) 

for  the  case  in  which  10  =0,  is  satisfied  at  all  points,  except  at  the  origin,  by 

z  =  Clr+c',  (72) 

where  C  and  c'  are  constants.  The  deflected  surface,  according  to  this 
equation,  is  a  surface  of  revolution  about  an  axis  through  the  origin.  A 
load,  concentrated  at  the  origin,  and  producing  the  state  of  flexure  defined 
by  (72),  may  be  called,  by  definition,  a  ring  load.  The  intensity  of  this 
ring  load  is  measured  conveniently  by  CEI,  where  El  is  the  usual  stiffness 
factor  of  the  slab.  Since  C  is  a  distance,  the  ring  load  CEI  may  be  meas- 
ured in  Ib.  in.3  units.  One  finds  by  differentiation  of  (72)  : 

^_  cy 

(73) 
*~x')       *L* ,  C(x_'-_y_v;      6jz        ZCxy 

<5yz          f4      '   <5x<5y          *"*  (74) 

that  is 

8zz.     <5Z2 

that  is  -=~?  +  v- /  »  Az  =  0  &nd    AAz  =•  0  . 

AV      Ay' 

According  to  formulas  (22)  in  Art.  6,  the  vertical  shears  are  proportional 
to  the  derivatives  of  &z;  that  is,  the  vertical  shears  are  zero  at  all  points 
except  the  origin.  The  moments  in  the  directions  of  x  and  y  are  defined  by 
the  second  derivatives  in  (74).  The  moment  in  the  direction  of  radius 
vector,  or  the  radial  moment,  is 

MJ-J  T    0   ~             O-/^/  /•7K\ 

—    —   £2    —    .  (  /  O  I 

In  a  circular  section  with  center  at  the  origin  and  with  radius  r,  there 
is,  then,  a  uniformly  distributed  radial  moment,  defined  by  (75),  but  no 
torsional  moment  and  no  vertical  shear.  In  the  light  of  the  state  of  flexure 
in  the  circular  section  with  center  at  the  origin  and  radius  r,,  one  may 
explain  the  nature  of  the  ring  load.  Assume  that  the  material  is  removed 
within  the  circle  with  radius  r,,  and  that  a  radial  moment,  determined  by 
(75)  is  applied  as  an  external  load,  uniformly  distributed  over  the  circum- 
ference of  the  circle.  Then  the  slab,  under  the  influence  of  this  load  alone. 


106  MOMENTS  AND  STRESSES  IN  SLABS. 

will  deflect  according  to  formula  (  72  )  ,  because  thereby  it  satisfies  all  the 
boundary  conditions.  Now  assume  that  the  circle  with  radius  r,  is  not  cut 
out,  but  that  instead  some  load  within  the  circle  produces  at  the  circumfer- 
ence of  the  circle  the  state  of  flexure  that  was  assumed  before  as  a  result 
of  the  external  loads.  On  account  of  the  identity  of  conditions  at  the  cir- 
cumference of  the  circle,  the  state  of  flexure  outside  the  circle  will  remain 
unchanged,  as  determined  by  equation  (72).  More  than  one  kind  of  load 
within  the  circle  may  produce  this  same  effect:  the  load  may  consist  of 
upward  and  downward  loads  ±  P,  uniformly  distributed  over  the  circumfer- 
ences or  areas  of  two  concentric  circles;  or  it  may  consist  of  an  upward 
load  P  at  the  origin  combined  with  a  down  load  P  which  is  uniformly  dis- 
tributed over  the  area  or  the  circumference  of  a  circle  with  center  at  the 
origin  and  radius  not  larger  than  r,.  But  whether  the  load  is  made  up  in 
one  way  or  another,  if  the  radius  rl  is  small,  the  load  may  be  considered  as 
one  concentrated  load;  namely,  the  ring  load  whose  magnitude  is  measured 
by  CEI,  and  whose  effects  are  defined  completely  by  equation  (  72  )  . 

It  may  be  noted  that,  according  to  equation  (72),  the  deflection  at  the 
origin  is  infinite.  Since  the  origin  lies  always  within  the  smallest  circle 
containing  the  whole  load,  the  infinite  deflection  at  the  center  has  only 
theoretical  significance.  When  the  ring  load  is  applied  at  a  point  of  sup- 
port, then,  in  order  to  avoid  the  assumption  of  any  infinite  deflections,  one 
may  conceive  of  the  support  as  being  distributed  over  the  circumference 
of  a  small  circle  whose  center  is  at  the  original  point  of  support  and  at  a 
fixed  elevation. 

The  expressions  (74)  are  well  suited  for  computations  of  such  series 
as  may  be  formed  when  a  large  number  of  ring  loads  are  applied  at  the 
same  time. 

(b)    Concentrated  couples. 

Lagrange's  equation   ((71)) 


for  w  =  0  is  satisfied  at  all  points  except  at  the  origin  by  the  solution 

z  =  A  x  I  .  r  +  ax,  (76) 

where  A   and  a  are  constants.     By  differentiation  one  finds: 


,77, 


&z  =  A—z  ,  (79) 

02 

this  value  of    A  ?  is  proportional  to  the  value  of   —  in  the  preceding  case, 

o£ 

(a),  which  gave    A^  =  0;    it  follows,  therefore,  in  the  present  case,  that 
AA  *  =  0. 

In  the  state  of  flexure  just  represented  the  ?/-axis  remains  undeflected. 
A  circle  drawn  on  the  slab,  with  center  at  the  origin  of  coordinates,  remains 


MOMENTS  AND  STRESSES  IN  SLABS.  107 

plane,  but  rotates  about  the  y-axis,  through  some  angle.  On  account  of  the 
anti-symmetry  with  respect  to  the  y-axis,  the  resultant  of  the  stresses 
in  the  cylindrical  section  r  =  r,  is  a  couple  about  the  y-axis.  Now  r, 
may  be  given  any  small  value,  that  is,  the  couple  must  be  transferred  to  the 
slab  at  the  origin  as  a  concentrated  couple. 

The  magnitude  of  the  concentrated  couple  may  be  found  by  considering 
the  stresses  in  two  sections  parallel  to  the  y-axis,  on  opposite  sides  of  the 
origin.  By  formulas  (22),  in  Art.  6,  one  finds  the  vertical  shear  per  unit- 
width  in  a  section  parallel  to  the  y-axis  to  be 


The  total  vertical  shear  in  this  section  becomes,  then, 


The  total  bending  moment  in  a  section  parallel  to  the  y-axis,  with  x  posi- 
tive, is  equal  to 


while  a  negative  x  gives  -)-  2^  AEI.  The  resultant  of  the  stresses  in  the 
two  sections  +  x  is  then  equal  to  the  concentrated  couple  =4  TT  AEI  (80), 
turning,  when  A.  is  positive,  in  the  direction  from  z  to  x. 

A  number  of  concentrated  couples  may  be  dealt  with  by  computing 
series  of  the  terms  contained  in  equations  (77)  and  (78). 

(c)   Ring  couples. 

The  deflection  due  to  two  equal  and  opposite  ring  loads,  close  to  the 
origin  and  to  one  another,  and  with  centers  on  the  x-axia,  may  be  expressed 
as  equal  to  a  constant  times  the  first  partial  derivative,  with  respect  to  x,  of 
the  deflection  which  is  due  to  a  single  ring  load,  and  is  expressed  by  equa- 
tion (72).  This  derivative  was  given  in  equation  (73).  Thus,  when  B  is 
a  constant,  the  function 

Z  =  *k,  (81) 

r2 

is  the  deflection  due  to  a  ring  couple  which  is  applied  at  the  origin  in  the 
direction  of  x,  and  is  measured  in  intensity  by  the  quantity  BEI.  One 
finds  by  differentiation  of  (81)  : 


£x  r4  Sy~         r*  (82) 

£7?  =  ~~^/  =  ~^r(/~  r2  !  '  (83) 

and  Az  =  AAz  =  0 


108 


MOMENTS  AND  STRESSES  IN  SLABS. 


(d)   Other  types  of  concentrated  loads. 

If  the  function  z  =  F(x,y)  satisfies  Lagrange's  equation  A  A  *  =  0 
at  all  points  except  at  the  origin,  which  is  a  singular  point,  then  also  the 
function 

gm+n  F 

2  "  8xm  8yn 

will  satisfy  the  equation  A  A*  =  0  at  all  points  except  at  the  origin ;  and 
z,  like  the  original  function  F,  will  define  some  concentrated  load  at  the 
origin.  Other  solutions  may  be  formed  by  integration  of  F.  Thus  from  the 
fundamental  solutions  (72)  and  (76),  for  ring  loads  and  concentrated 
couples,  and  from  the  solution 


rl.rdr 


(85) 


which  defines  a  single  concentrated  force  proportional  to  D,,  at  the  origin, 
one  may  derive  an  infinite  number  of  solutions,  each  defining  a  correspond- 


y 

Y 

R, 

>-     -t 

**  1 

J 
^                  -« 

?m 
>-         -« 

, 

^       < 

J  1 

-z- 

—  J;  - 

-z* 

—     I     -» 

~l- 

-z* 

^z^ 

N 

, 

1 

FIG.  A2. — RING  LOADS  IN  A  Row. 


ing  concentrated  load.  It  is  an  example  of  this  procedure,  that  the  deflec- 
tions due  to  the  ring  couple  were  derived  by  differentiation  of  the  expression 
for  the  deflections  due  to  the  ring  load. 

(e)    The  co-action  of  a  number  of  ring  loads. 

A  number  of  equal  ring  loads,  R^ 7?n,  of  intensity  CEI,  is 

assumed  to  act  in  a  row  parallel  to  the  #-axis  as  shown  in  Fig.  A2.  The 
spacing  is  constant  and  equal  to  I.  By  using  equations  (73)  and  the  rela- 
tions £  =  x  4.  ,,  one  finds  the  change  of  slope  between  the  two  origins  of 
coordinates,  o  and  a  at  a  distance  of  I,  to  be 


(86) 

that  is,  the  difference  is  expressed  in  terms  of  the  coordinates  of  the  first 
and  the  last  load  in  the  row. 

In  Fig.  A3  ring  loads  of  intensity  CEI  are  applied  at  the  points  marked 
with  small  circles.  The  group  of  loads  is  symmetrical  with  respect  to  the 
lines  y  =  Q,  a='1/2l,  and  a?  =1&l  =*=  y.  According  to  (86)  the  change  of 
slope  between  the  points  a  and  6  may  be  expressed  as  follows,  in  terms  of 


MOMENTS  AND  STRESSES  IN  SLABS. 


109 


the  coordinates  of  the  extreme  loads  in  the  first  quadrant,  that  is,  the  loads 
on  the  line  A : 

S-'.^C  AP-  (87) 

If  the  number  of  loads  is  very  large,  then  the  summation  may  be  replaced 
by  an  integration.    By  making  use  of  the  axes  of  symmetry,  one  finds  then 


r 

x,     <dy     2"C  f  *dy~  ydx    2.C.  /" *  r  08    irC, 


This  difference  of  slope  may  be  reduced  to  zero  if  at  the  edge  of  the  plate, 
which  is  assumed  to  be  at  infinity,  a  certain  additional  load  is  applied; 


FIG.  A3. — GKOUP  OP  RING  LOADS. 


namely,  a  bending  moment,  uniformly  distributed  over  the  edge,  producing 
a  uniform  bending  moment  in  the  slab  in  all  directions,  equal  to 

(89) 


With  this  additional  moment  present,  small  circles,  drawn  in  the  central 
portion  of  the  slab,  concentric  with  the  ring  loads,  all  with  the  same  radius, 
will  remain  co-planar;  and  they  will  be  contour  lines  of  the  deflected 
surface  if  one  of  them  remains  horizontal.  On  account  of  this  relation  to 
the  ring  loads  the  moment  M  is  assumed,  in  the  analysis  of  the  uniformly 
loaded  flat  slab,  to  act  with  the  group  of  ring  loads. 

(f)    The  co-action  of  concentrated  couples  and  ring  couples. 

A  concentrated  couple  4  T  AEI,  and  a  ring  couple  BEI,  whose  effects 
are  determined  by  equations  (76)  and  (81),  respectively,  are  assumed  to 
act  on  the  slab,  at  the  origin,  in  the  o^-plane.  The  circles  r-=  const,  remain 
plane  under  the  influence  of  this  combined  load.  The  slope  in  the  direction 


110 


MOMENTS  AND  STRESSES  IN  SLABS. 


of  x  varies,  in  general,  from  one  point  to  another  on  the  circle  r  =  const. 
The  condition  may  be  imposed,  however,  that  all  points  of  a  certain  circle, 

for  example,  the  circle  r=  *L    which  marks  the  edge  of  a  column  capital, 

Jt 
must  have  a  common  tangential  plane.    By  comparing  equations  (77)  and 

(82)  one  finds  that  all  elements  of  the  deflected  surface  at  the  circle  r  =   — 


f™ 


I- 


C-rB-p— A — r&T C rB-r-A — r^n C -rBn — A- 


~  :~c 


r-X 

cc 


np 

CD 


(0) 


5-  •  25  q'l 

«T)JD 


X-~N 


za 


B-1 


5-  -25-1'  b 


=54- 


2-  •  25-1'b 


C  Jfj 


p* 


5  V 

ii:o'b 


a 


i-db 


[5V 

-i 


-1- 


—  1 


-H 


I £'- 


FIG.  Bl.^ — REINFORCING  PLAN  FOB  SLAB  J. 


have  a  zero  slope  in  the  direction  of  y,  and  consequently    have  a  common 
tangential  plane,  when 

B=4--  (90) 


Concentrated  couples     =*»  ^TtAEI    and  ring  couples    =t  B£7    are  used 
in  the  analysis  of  flat  slabs  with  alternate  rows  of  panels  unloaded.     The 


MOMENTS  AND  STRESSES  IN  SLABS. 


Ill 


rows  considered  are  parallel  to  the  i/-axis.  One  concentrated  load  of  each 
kind  is  assumed  at  each  column  center.  The  double  signs  =t  refer  to 
alternate  rows  of  columns.  Because  of  the  loads  at  the  surrounding  sup- 
ports, equation  (90)  expresses  in  this  case  only  approximately  the  condi- 
tion that  there  is  a  common  tangential  plane  at  all  points  of  the  circle 

r  =  —  .     The  following  formula,  which*  is  a  modified  form  of    (90),  and 

£ 

which  is  a  close  approximation,  takes  into  consideration  the  concentrated 


©    © 


©©  ©  ©@©  ©  © 

FIG.  E2. — I  OCATION  OF  GAGE  IINES  FOR  TOP  OF  FLAB  S. 


loads  at  the  nine  points  definea  by  the  coordinates  y  =  —  I,  0,  -f-  I,  and  cc 
=  ±  /  (negative  values  of  A  and  B)  and  x  =  0  (positive  A  and  B)  ;this 
formula  was  derived  by  equating  to  one  another  the  slopes  in  the  direction 


of  x  at  the  points  /O,  -\   and  (-  ,  0\  : 


(91) 


Formula   (91)   was  used  in  computing  values  of  B  when  alternate  rows  of 
flat  slab  panels  are  unloaded. 


112  MOMENTS  AND  STRESSES  IN  SLABS. 

APPENDIX    B. 

TESTS  OF  SLABS  AT  PURDUE  UNIVERSITY 
BY  W.  A.  SLATER. 

Bl.  Description  of  Tests.     The  tests  referred  to  as  the  Purdue  teata 
were  made  for  the  Corrugated  Bar.  Co.  under  the  direction  of  Prof.  W.  K. 


flfeZE 

Gaqe  lines  are  shawoi 
they  mutt  qfeor  if  seen 
through  the  slab  from  obnv 


FIG.  B3. — LOCATION  OF  GAGE  LINES  FOB  BOTTOM  OF  SLAB  J. 

Hatt  at  Purdue  University,  Lafayette,  Ind.,  on  two  test  slabs,  J  and  S, 
each  of  which  had  four  panels  16  ft.  square. 

The  dimensions  of  the  concrete  in  the  two  slabs  were  the  same,  but  the 


MOMENTS  AND  STRESSES  IN  SLABS. 


113 


Goqe  lines  are  shown 
a5  they  would  appear  if 


seen  through  the  column 
from  the  outside 


FIG.  B4. — LOCATION  OF  GAGE  LINES  ON  COLUMNS  AND  MARGINAL  BEAMS 

OF  SLAB  J. 


Unit  Be*mutKn  ardy^s  to  Scale  Mealed          J-awH  oammoxmim 

SmriySfpff^n 

FIG.  B5. — LOAD  STRAIN  AND  LOAD-STRESS  DIAGRAMS  FOR  Top  OF  SLAB  J. 


114 


MOMENTS  AND  STRESSES  IN  SLABS. 


- 


MOMENTS  AND  STRESSES  IN  SLABS.  115 

amount  of  reinforcement  for  slab  S  was  considerably  less  than  that  for  slab  J. 
The  latter  fact  will  help  to  account  for  the  smaller  load  which  was  carried 
by  slab  S  than  by  slab  J,  but  another  important  consideration  is  the  fact 
that  for  slab  S  the  average  strength  of  the  concrete  control  cylinders  at  28 
days  was  only  1215  Ib.  per  sq.  in.  while  the  strength  of  the  control  cylinders 
for  slab  J  was  2305  Ib.  per  sq.  in. 

The  methods  of  making  the  test  are  similar  to  those  which  have  been 
described  in  reports  of  various  tests  on  floors  of  buildings.* 

The  important  dimensions  of  the  slab  are  shown  in  Table  XIII.  The 
amount  and  distribution  of  the  reinforcement  are  shown  in  Fig.  Bl  and  B9. 


0    JO     0     10     0     JO     0    .10     0     10    ?0   30  40    50   60    70    60    00  100  I/O  120  130  1.40  ISO  160 
Deflection.  Inches 

Fio.  B7. — LOAD  DEFLECTION  DIAGRAMS  FOR  SLAB  J. 

The  location  of  gage  lines  is  shown  in  Fig.  B:>,  B3,  B4,  BIO,  Bll,  and  B12. 
The  measured  stresses  in  the  reinforcement  and  deformation  in  the  concrete 
are  shown  in  Fig.  B5,  BO,  B13  and  B14.  The  deflections  are  shown  in  Fig. 
B7  and  B8.  Certain  information  concerning  these  tests  has  already  been 
published!  and  reference  to  the  published  report  will  supply  certain  results 
of  the  test  which  are  lacking  in  this  paper. 

In  the  tests  of  both  slabs  the  load  Avas  applied  as  nearly  uniformly  as 
possible.  In  order  to  afford  access  to  the  gage  lines  on  the  top  surface  of 
the  slabs,  aisles  were  left  in  the  loaded  area.  When  the  load  was  high 
enough  these  aisles  were  bridged  over  and  sufficient  load  was  placed 
immediately  over  them  to  give  practically  a  uniform  distribution  of  load. 


*  Univ.  of  111.   Eng.   Exper.   Sta.   Bulletins  64  and   84. 

t  W    K    Halt     "Moment  Coefficients   for  Flat-slab  Design   with   Results  of  a  Test," 
Froc.  A.  C.  I.   V.   14,  p.   174   (1918). 


116 


MOMENTS  AND  STRESSES  IN  SLABS. 


B2.  Loading  of  Slab  J.  In  the  test  of  slab  J  the  highest  load  applied 
uniformly  over  the  entire  slab  was  595  Ib.  per  sq.  ft.  At  this  load  the 
measured  stress  in  the  reinforcement  was  at  the  yield  point  in  gage  lines 
which  crossed  the  mid-section  of  the  slab  (see  Fig.  12  which  shows  location 
of  sections),  and  the  highest  deflection  reported  for  any  panel  was  1.1  in. 
After  the  load  had  been  in  place  about  two  days  longer  the  deflection  had 
increased  to  1.25  in.  At  this  stage  of  the  test  it  is  reported  that  there  was 
no  evidence  of  crushing  of  the  concrete.  The  entire  load  was  removed  from 
the  slab  and  about  40  days  later  a  load  of  803  Ib.  per  sq.  ft.  was  applied 
"over  one  panel,  the  overhang,  and  into  the  adjoining  panel,  etc."*  Failure 
occurred  under  this  load  by  punching  of  the  column*  capital  through  the 


0016 


Legend 

Section  A- A  inferior  panel 
•     B-B  wall       •• 


FIG.  B8.  —  STRAIN  DISTRIBUTION  FOR  REINFORCEMENT  IN  WALL  PANEL  AND 
INTERIOR  PANEL  OF  SLAB  J. 


dropped  panel.  The  fracture  had  the  angle  of  a  diagonal  tension  failure. 
Assuming  the  full  live  and  dead  load  of  an  area  16  ft.  square  to  have 
been  carried  on  the  central  column,  the  computed  shearing  stress  on  the 
vertical  section  of  depth  jd,  which  lies  at  a  distance  d  from  the  edge  of  the 
column  capital,  was  233  Ib.  per  sq.  in.  Although  the  "failure  of  the  slab 
.  .  .  began  with  a  feathering  of  the  concrete  on  the  dropped  panel  at  the 
edge  of  the  column  capital"*  this  shearing  stress  is  high  enough  that  it 
seems  that  diagonal  tension  may  have  been  a  factor  in  causing  failure. 

B3.  Loading  of  Slab  S.  The  maximum  load  applied  to  slab  S  was 
450  Ib.  per  sq.  ft.  The  official  report  of  the  test  states  that  this  load  "was 
attended  by  complete  failure  of  the  concrete  in  compression  and  the  stretch- 
ing of  the  steel  to  the  yield  point."*  The  load-strain  diagrams,  Fig.  B13 
and  B14,  show  that  the  reinforcement  generally  was  highly  stressed  both 
at  sections  of  negative  moment  and  at  sections  of  positive  moment,  and 


•Proceedings  A.  C.  I.,  Vol.  XIV,  pp.  182  and  183   (1918). 


MOMENTS  AND  STRESSES  IN  SLABS. 


117 


photographs  of  the  slabs  show  the  crushing  of  the  concrete  around  the 
capital.  However,  the  highest  deflection  reported  was  only  1.30  in.  at  the 
center  of  a  panel,  and  when  the  load  was  removed  the  deflection  decreased 
to  0.4  in.  Relatively  this  deflection  was  small  and  the  recovery  was  large 
and  the  test  does  not  afford  a  conclusive  answer  to  the  question  as  to  what 
load  would  have  been  required  to  cause  collapse  of  the  structure,  or,  in  other 
words,  as  to  what  was  the  factor  of  safety  against  destruction  of  life  and 


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FIG.  B9. — REINFORCING  PLANS  FOB  SLAB  S. 

property.  In  view  of  the  large  load  carried  by  the  Waynesburg  slab  after 
the  yield  point  of  the  negative  reinforcement  had  been  reached  it  does  not 
seem  unreasonable  to  believe  that  this  slab  might  have  carried  more  load 
without  actual  collapse. 

B4.  Moments  in  Wall  Panels.  In  the  design  of  slab  J  and  slab  S 
provision  for  greater  positive  moments  in  the  wall  panels  than  in  the 
interior  panels  was  made  by  using  a  larger  area  of  reinforcement  for  the 
positive  moment  in  the  wall  panels  than  in  the  interior  panels.  The  same 


118 


MOMENTS  AND  STRESSES  IN  SLABS. 


number  of  bars  was  used  at  the  two  positions,  but  for  the  wall  panels  square 
bars  were  used  and  for  the  interior  panels  round  bars  were  used.  This 
gives  27  per  cent  more  reinforcement  for  the  wall  panel  than  for  the  interior 
panel.  Strains  measured  are  shown  in  Fig.  B8  and  B16.  For  the  lower 
loads  the  stresses  were  almost  equal  in  the  two  panels  of  slab  J,  but  were 
somewhat  higher  for  the  wall  panel  in  slab  S  than  for  the  interior  panel. 
For  the  higher  loads  the  stresses  were  higher  for  the  interior  panels  in 


FIG.  BIO.— LOCATION  OF  GAGE  LINES  FOB  Top  OF  SLAB  S. 

both  cases.  The  indication  from  this  test  is  that  the  allowance  of  27  per 
cent  greater  moment  for  wall  panels  than  for  interior  panels  was  in  excess 
of  the  requirement  for  wall  panels.  The  moments  in  the  wall  panels  will  be 
dependent  upon  the  moment  of  inertia  of  the  wall  columns  and  probably  upon 
the  manner  in  which  the  negative  reinforcement  at  the  edge  of  the  wall 
panel  is  distributed,  and  for  this  reason  the  results  in  Fig.  B8  and  B16 
should  not  be  applied  to  other  cases  without  taking  into  account  the  effect 
of  these  features  of  the  design. 


MOMENTS  AND  STRESSES  IN  SLABS. 


119 


APPENDIX  C. 
BIBLIOGRAPHY. 

References  are  made  in  the  following  list,  to  published  results  and  to 
some  unpublished  results  of  tests  on  flat  slabs  or  on  slabs  supported  on 
beams  which  lie  on  the  edges  of  the  panels,  but  which  have  no  intermediate 
beams. 

In  general  the  following  sequence  is  used  in  references  cited:  Name  or 
designation  of  structure  tested,  city,  brief  characterization  of  type  of  rein- 
forcement, number  of  panels  loaded,  reference  to  periodicals  by  number  in 
parentheses,  date  of  publication. 


NDTE--Z  qxf.  foBoO  on  N  side 
of  this  column  txbn  cap 


f  -  Oaqe  lines  are  tfom  as  they  nou/d appear 
if  seen  through  ffe  shb  from,etwe. 


FIG.  Bll. — LOCATION  OF  GAGE  LINES  FOR  BOTTOM  OF  SLAB  S. 


—  -^^JT^VTPX^C*  •  :  ~^''ft=&MMy3<gy3?*$c  "T 
...^^^^.^^^j.-              -.f  t-^-^^V^^S^^—  -' 

..„_..., 

Gaqe  lines  on  inner  face 

NOTE.  :  Gaqe  lines  are  shown 

ssXL, 
<3#rw 

as  they  would  appear  if 

seen  through  the  column 

from  the  ot/fyde. 

|                                       | 

Fro.  B12. — LOCATION  OF  GAGE  LINES  ox  COLUMN  Z  AND  MARGINAL  BEAKS 

07  SLAB  S. 


120 


MOMENTS  AND  STRESSES  IN  SLABS. 


The  periodicals  or   institutions   referred   to  in   the  bibliography  are 
designated  by  the  following  numbers: 

( 1 )  Proceedings   National   Association   of   Cement  Users   and   of   its 

successor,  the  American  Concrete  Institute. 

(2)  University  of  Illinois,  Engineering  Experiment  Station. 

(3)  Indiana  Engineering  Society. 

(4)  Proceedings  Pacific  Northwest  Society  of  Civil  Engineers. 


•  Gaqe  lines  on  steel 
•      •     •  concrete 

FIG.  B13. — LOAD  STRAIN  AND  LOAD-STRESS  DIAGRAMS  FOR  TOP  OF  SLAB  S. 


(5)  Transactions  American  Society  of  Civil  Engineers. 

(6)  Journal  of  the  Engineering  Institute  of  Canada. 

(7)  Bulletin  on  Flat  Slabs  by  Corrugated  Bar  Co. 

(8)  Engineering  and  Contracting. 

(9)  Engineering  News. 

(10)  Engineering  Record. 

(11)  American  Architect. 


MOMENTS  AND  STRESSES  IN  SLABS. 


121 


122 


MOMENTS  AND  STRESSES  IN  SLABS. 


.10    .29    0     10    ?0   30   40  .X  .60    W    BO    90  ICC  110  12)  130  140 

Deflection,  Inches, 
FIG.  B15. — LOAD  DEFLECTION  DIAGRAMS  FOB  SLAB  S. 


Legend 

5ecttonA-A  interior  panel 
•     B-B  wall 


•^TX?X*K7*^ 


,f .-•/.  - .. 


FIG.   B16. — STRAIN  DISTRIBUTION  IN  WALL  PANEL  AND  INTERIOR   PANEL 

OF  SLAB  S. 


MOMENTS  AND  STRESSES  IN  SLABS.  123 

LIST    OF    TESTS. 

1.  C.  Bach  and  O.  Graf:    Versuche  mit  allseitig  aufliegenden,  quadra- 
tischen  and  rechteckigen  Eisenbetonplatten,  Deutscher  Ausschuss  fiir  Eisen- 
beton,  v.  30,  Berlin,  1915,  309  pp.     These  laboratory  testa  were  made  in 
Stuttgart,  1911  to  1914,  under  the  direction  of  Bach  and  Graf.     52  slabs 
supported  on  four  sides  and  35  control  strips  supported  as  beams  were 
tested  to  failure.     The  tests  are  reported  in  detail,  without  attempt,  how- 
ever, to  explain  or  analyze  the  results.     Analyses  of  the  results  have  been 
made  later  by  Suenson  and   by  Nielsen;     see  E.   Suenson,  Krydsarmerede 
Jaernbetonpladers   Styrke,   Ingenioeren    (Copenhagen),    1916,   No.    76,   77, 
and   78;     N.   J.   Nielsen,   Krydsarmerede   Jaernbetonpladers   Styrke,   Inge- 
nioeren, 1920,  pp.  723-728. 

2.  Deere   and   Webber   Building,   Minneapolis,    1910,   4-way,   9   panels, 
(1)   1910,  (2)   Bull.  64,  1911,  (9)   12-22-1910,   (8)   12-22-1910. 

3.  Test  of  Rubber  Model  Flat  Slab,   1911,  9   panels,    (7)    1912,    (1) 
1912,  p.  219. 

4.  Powers  Building,  Minneapolis,   1911,  2-way,  4  panels,    (1)    1912,  p. 
61,   (9)   4-18-1912,   (10)   4-20-1912. 

5.  Franks  Building,  Chicago,  1911,  4-way,  4  panels,  (1)   1912,  p.  160. 

6.  Barr   Building   Test   Panel,    St.    Louis,    1911,    2-way    supported   on 
beams,  (1)   1912,  p.  133. 

7.  St.  Paul  Bread  Co.  Bldg.,  1912,  4-way,  1  panel,  (5)   1914,  p.  1376. 

8.  Larkin   Building,   Chicago,   1912,   4-way,  5   panels,    (1)    1913,    (10) 
1-1913. 

9.  Northwestern   Glass   Company   Building,  Minneapolis,   1913,   4-way, 
4  panels,  (5)   1914,  p.  1340. 

10.  Worcester  (Mass.)  Test  Slab,  1913,  (2)  Bull.  84,  1916. 

11.  Shredded  Wheat   Factory,  Niagara  Falls,  N.  Y.,    1913,  2-way,   9 
panels,  (1)    1914,   (2)   Bull.  84,  1916. 

12.  International  Hall,  Chicago,  1913,  4-way,  4  panels,    (5)    1914,  pp. 
1433-1437. 

13.  Soo  Line  Terminal,  Chicago,  1913,  4-way,  4  and  5  panels,  (2)   Bull. 
84,  1916,  (9)   8-16-1913. 

14.  Curtis  Ledger  Factory,  Chicago,   1913,  2-way  at   columns,  4-way 
elsewhere,  4  panels,   (2)    Bull.  84,  1916. 

15.  Schulze  Baking  Co.  Building,  Chicago,  1914,  4-way,  4  panels,   (2) 
Bull.  84,  1916. 

16.  Schwinn   Building,    1914-15,   long   time   test,   4-way,    1    panel,    (1) 
1917,  p.  45. 

17.  Sears  Roebuck  Building,  Seattle,  1915,  2-way,    (4)    Jan.  and  Feb., 
1916. 

18.  Bell  St.  Warehouse,  Seattle,  1915,  4-way,  4  panels,   (4)    1916,   (10) 
5-13-16. 

19.  Eaton   Factory.  Toronto,   Out.,   about    1016.   4-way,   4   panels,    (6) 
April,  1919. 


124  MOMENTS  AND  STRESSES  IN  SLABS. 

20.  S.-M.-I  Slab,  Purdue  University    (1917)    circumferential  reinf.,  4 
panels,  (1)   1918. 

21.  Sanitary  Can  Building,  Maywood,  111.,  1917,  2-way,  4  panels,  (1) 
1917,  p.  172,  and  1921,  p.  500. 

22.  Shonk  Building,  Maywood,  111.,  1917,  4-way,  4  panels,  (1)   1917,  p. 
172,  and  1921,  p.  500. 

23.  Western  Newspaper  Union  Building,  Chicago,  1917,  4-way,  4  panels, 
(1)   1918,  p.  291,  (2)  Bulletin  106,  1918. 

24.  Slabs  J  and  S,  Purdue  Univ.,  1917,  2-way,  4  panels  each,  (1)   1918, 
p.  174,  also  1921,  p.  500. 

25.  Slab  R,  Purdue  Univ.,  1917,  circumferential"  reinf.,  4  panels,   (1) 
1917,  p.  172. 

26.  Arlington  Building,  Washington,  D.  C.,  1918,  2-way  tile  and  con- 
crete supported  on  beams.      (Under  preparation  as  Tech.  paper  of  U.  S. 
Bureau  of  Standards.) 

27.  Whitacre    Test    Slab,    Waynesburg,    Ohio,    1920,    2-way,    tile    and 
concrete,  supported  on  beams,  18  panels,  (11)  8-11-20  and  3-16-21  also  under 
preparation  as  Tech.  Paper  U.  S.  Bureau  of  Standards. 

28.  Channon  Building,  Chicago,  1920,  circumferential  reinf.  4  panels, 
(1)  1921,  p.  500. 

29.  Jersey  City  Dairy  Company's  Building,  Jersey  City,  N".  J.,  1913, 
2-way,  1  panel,  tested  by  Corrugated  Bar  Co.,  Buffalo,  N.  Y.,  not  published. 


